Did Jesus exist?
We have to be very careful in phrasing this question. If we ask - "Did Jesus - you know, the guy bescribed in the Bible - really exist?" Then we are forced to reject this as a question of faith, since the guy in the gospels is described as born by a virgin, resurrected from the dead and risen to the heavens etc. Even without examining the supernatural claims, the gospels contain internal contradictions which means they cannot possibly be 100% true.
But we can pose the question in a different way so we can actually get a meaningful answer.
Lets first state what we know beyound a doubt: We know that a new religious movement originated among Jews in first century Palestine and from there spread out over the roman empire. These people believed that a man from Galilee were the Messiah and the Son of God, and that his preaching and exploits are described in the Gospels. We know that the gospels were written and compiled during the first and second century.
So the question becomes: Is the narrative of the Jesus figure in the gospels originally based on a real person who lived in first century Palestine and which instigated the movement, or is he a purely fictional or mythical construction?
Realize that we cannot prove either hypothesis.
We have no first-hand or contemporary sources to the life of Jesus and we have no archaeological remains to prove his existence. And this is not surprising - this is the case for the vast majority of pre-modern historical figures, including people who was much more famous in their own time than Jesus. Take someone like Alexander the Great, one of the most important figures in History. We have no first-hand sources and no remains. The best sources we have are written hundreds of years after his death. Even then, given the available historical evidence we assume Alexander existed, since a lot history is impossible to explain otherwise (and his invention would have required a vast conspiracy among historians in antiquity).
The outlandish claims regarding Jesus (walked on water, resurrected the dead etc.) cannot be taken as evidence that he is fictional. If we compare Alexander again, multiple legends grew up around him - he was the son of Zeus, his sister was a mermaid etc. This just shows that when a figure becomes revered, legends will form. No historians in their right mind would deny that Alexander was historical. Stripping away all the legends around Alexander there is still a historical core which is more likely to be true than to be made up. I'm going to explain why historians consider the same to be the case for Jesus.
A note on sources. You explicitly ask for non-christian sources. Ill go ahead and ignore this, because I suspect it is based on the common but mistaken assumptions that the christian sources (i.e. the Gospels) are inherently more unreliable than other sources. Sure, the bible is biased and makes outrageous claims about the miracles and divinity of Jesus - but be aware that all ancient sources are unashamedly biased and filled with with outrageous claims by modern standards.
Take Josephus. He is considered a serious historian, but starts out his history book with:
In the beginning God created the heaven and the earth. But when the
earth did not come into sight, but was covered with thick darkness,
and a wind moved upon its surface, God commanded that there should be
light: and when that was made, he considered the whole mass, and
separated the light and the darkness; (Chapter 1)
...he goes on to describe Adam and Eve, the ark of Noah and so on. Even then Josephus is considered one of the most important and reliable historical sources. We just have to read it all with a critical mind.
Or take Tacitus, considered "the pinnacle of Roman historical writing". He writes about Christianity:
"Christus, from whom the name had its origin, suffered the extreme
penalty during the reign of Tiberius at the hands of one of our
procurators, Pontius Pilatus, and a most mischievous superstition,
thus checked for the moment, again broke out not only in Judæa, the
first source of the evil, but even in Rome, where all things hideous
and shameful from every part of the world find their centre and become
Clearly he is at least as biased as the Christians, just with the opposite bent!
He treats Saturn, Jupiter and Isis as real in his Histories, and about Vespasians siege of Jerusalem he writes:
Prodigies had occurred, which this nation, prone to superstition, but
hating all religious rites, did not deem it lawful to expiate by
offering and sacrifice. There had been seen hosts joining battle in
the skies, the fiery gleam of arms, the temple illuminated by a sudden
radiance from the clouds. The doors of the inner shrine were suddenly
thrown open, and a voice of more than mortal tone was heard to cry
that the Gods were departing. At the same instant there was a mighty
stir as of departure. Some few put a fearful meaning on these events,
but in most there was a firm persuasion, that in the ancient records
of their priests was contained a prediction of how at this very time
the East was to grow powerful, and rulers, coming from Judaea, were to
acquire universal empire. These mysterious prophecies had pointed to
Vespasian and Titus, but the common people, with the usual blindness
of ambition, had interpreted these mighty destinies of themselves, and
could not be brought even by disasters to believe the truth. (Histories 18)
Here Tacitus mentions a number of supernatural occurrences just as amazing as what is described in the gospels AND basically claims that Vespasian (the roman emperor) is really the messiah, and anybody who cannot see this is an idiot! And despite passages like this he is still considered one of the most reliable ancient historians...
Bottom line: Any ancient text should be read with a heavy dose of source-criticism. The gospels are actually relatively strong sources because we have four semi-independent account. The inconsistencies between the gospels have been a source of embarrassment for the church, but for the historian they are a godsend: They allow us to through textual analysis determine the age of various parts of the narrative and then estimate how it developed over time. The gospels taken together are actually pretty good source material compared to what we have about most other pre-modern figures, including kings and emperors, believe it or not.
Now if we take the gospels and strip away anything slightly supernatural, we are left with a pretty plausible narrative. A Jewish guy walks around in Galilee and Judea and talks in parables and preaches about the coming Kingdom of God. He gains a bunch of followers (a cult would we probably say today) and is rumored to be the long awaited Messiah. The Romans consider this cult a threat, and execute the leader. Not only is this pretty plausible, it actually happened more or less like this several times. The historian Josephus describes several such "mad prophets" and even a similar episode where a guy claiming to be the messiah gets a bunch of followers, leads them into the desert where he claims to want to part the Jordan river (like Moses parted the Red Sea) ... but before he gets that far is slaughtered by Roman soldiers along with his followers.
Whats more, the Romans actually acted rationally (if cynically) by killing anybody claiming to be the messiah before the cult became to strong, For example the bloody Bar Kokbha revolt was instigated by a rebel-leader claiming to be the Messiah. (Though reading the gospels critically it is not totally clear if Jesus considered himself the Messiah, but what is clear is that some of his followers did, which was enough that the Romans considered him dangerous.)
The only really unique thing about Jesus compared to these other prophets and cult-leaders, is that his cult survived his death and turned into a major religion in the Roman Empire. But this seems to be more due to Paul, which transformed an obscure Jewish sect into a religion acceptable to non-Jews and started missioning among the polytheists in the Roman empire, at at time that were ripe for Monotheism.
Now lets examine the other hypothesis, that Jesus is a purely fictional or mythological figure. Even if that is the case, someone wrote (or composed, since it was initially an oral tradition) the sermon on the mount, the parables and so on, and created the narrative of his ministry, followers, crucifixion etc. Lets call this hypothetical person or group of persons Non-Jesus.
There are some urban legends circulating in dark corners of the internet about parallels between Jesus and the Horus myth or the Bacchus myth or some other myths which is supposed to the be "true origin" of the Jesus myth. Forget about this. Sure, there a few parallels across religion (ie. resurrection is a recurring theme, miracles surrounding the birth of demigods etc), but the vast majority or the religious content of the gospels are very specific Jewish culture. There are numerous references to mosaic law and the OT prophets, discussions about interpretations of the laws and tradition, references to temple cult, references to groups like Pharisees and Sadukees and references to the
roman occupation. If Jesus is fictional, Non-Jesus was without a doubt a first century Jew.
However if we look at it as a work of fiction it has some strange strange choices. If the purpose of the narrative is to show that Jesus was the Messiah / the Son of God, then why invent the story that he was crucified? Why don't say that he killed a thousand Romans with a flaming sword and then was lifted into the heavens by the hand of God or something like that?
This is the criterion of embarrassment. It states that if some account in the gospels is embarrassing to the Christians, they probably didn't make it up themselves! The crucifixion is the clearest example of this principle because this was absolutely not supposed to happen, and it did in no way help the Christians convince others that Jesus was the Messiah. (Note that the Messiah at this time was supposed to literally be a triumphant king beating the enemies and restoring the kingdom to its former glory. It was only after Christianity re-interpreted the prophecies we got the idea of the "spiritual messiah".)
If we examine the gospels chronologically we have in the oldest version, Mark:
At noon, darkness came over the whole land until three in the
afternoon. And at three in the afternoon Jesus cried out in a loud
voice, “Eloi, Eloi, lema sabachthani?” (which means “My God, my God,
why have you forsaken me?”). When some of those standing near
heard this, they said, “Listen, he’s calling Elijah.” Someone ran,
filled a sponge with wine vinegar, put it on a staff, and offered it
to Jesus to drink. “Now leave him alone. Let’s see if Elijah comes to
take him down,” he said. With a loud cry, Jesus breathed his last.
This is extremely embarrassing. In the last moment he looses faith, thinks God has abandoned him, and dies with a whimper. Even Jim Jones or David Koresh did not lose faith like this! And people around him does not understand what he is saying and are making fun of him.
Now consider the later account in Luke:
It was now about noon, and darkness came over the whole land until
three in the afternoon, for the sun stopped shining. And the
curtain of the temple was torn in two. Jesus called out with a loud
voice, “Father, into your hands I commit my spirit.” When he had
said this, he breathed his last. Luke 20
Now the story is less embarrassing. No crying or loosing faith. Instead Jesus bravely accepts his fate.
Now see the latest version in John:
Later, knowing that everything had now been finished, and so that
Scripture would be fulfilled, Jesus said, “I am thirsty.” A jar of
wine vinegar was there, so they soaked a sponge in it, put the sponge
on a stalk of the hyssop plant, and lifted it to Jesus’ lips. When
he had received the drink, Jesus said, “It is finished.” With that, he
bowed his head and gave up his spirit. John 19
Now, not only does Jesus bravely accept his fate, it actually turns out it was part of the big plan all along! And the sponge-guy is not taunting Jesus anymore, but rather offering the sponge as an act of grace. The whole scene is just so much more dignified and meaningful.
It is quite easy to see how the narrative could have developed like this. First the crucifixion and the pitiful death of Jesus is an embarrassing fact. But over the following decades (it is assumed to be about 40 years between Mark and John) it is re-interpreted (or spun as we would say today) to actually becoming a meaningful event which shows the greatness of Jesus, and was planned all along.
Now consider if it was a fictional account, made up by Non-Jesus. This makes no sense to make up such an embarrassing end for Jesus (which the Christians later had to spin heavily), if the point of the story was to convince people that this guy was God! The case for the purely fictional Jesus simply does not make sense.
Tl;dr: Jesus most likely were a real person.
Bart D. Ehrman; Did Jesus Exist?: The Historical Argument for Jesus of Nazareth is one book which draws the same conclusion (but in more detail), but what I'm stating is not controversial. The Historicity of Jesus page states:
There is "near universal consensus" among scholars that Jesus existed
And provides numerous references. The "Jesus was a myth"-theory is basically a fringe theory at this point.
What is still controversial is exactly how much of the gospels actually happened and how much is later legends. This is an ongoing discussion, and a far more complex question.