My favorite podcast dedicated a rather long and thorough segment to this a while back in 2013-03-16. I think the result was that this could not be reproduced in studies.
Although the "both eyes at once" bit is a twist I hadn't heard of before today.
Podcast link: http://www.trcpodcast.com/trc-236-crying-ghost-orbs-napoleon/
(Cross link: same topic on Snopes: Crying left and right)
Edit 2014-11-25: Transcript from podcast episode below.
[05:55/38:24] Pat Roach:
Doctor Nolfi is a well respected optometrist and recognised lecturer within the vision care industry. He’s the co-founder of the Toronto eye care clinic […].
[06:20/38:24] Pat Roach:
So I wrote Jerry a note […] I think you’ll find his response interesting. […]
[07:15/38:24] Pat Roach quoting Jerry Nolfi:
Based on my knowledge of the tearing system there appears to be no logical reason for each eye to discriminate its tearing function based on the emotional stressor/stimuli. I just don’t see it and the statement must be a myth.
[08:10/38:24] Pat Roach:
What I really needed was an expert on crying. And I actually found one.
[08:20/38:24] Pat Roach:
So doctor “Ad Vingerhoets”
is a psychology professor at Tilburg University in the Netherlands. He’s one of the few psychologists in the world to actually have studied crying. […]
[10:55/38:24] Pat Roach:
He had videos of people starting to cry and he was willing to have students analyse whether or not it was the left or the right eye based on the specific stimuli.
[11:04/38:24] Pat Roach quoting Dr. Ad Vingerhoets:
Here’s a report about our observations.
Ten participants cried during a “Loss” scene while watching a movie. And three participants cried during a “Reunion” scene.
What was the movie?
He didn’t mention.
[11:21/38:24] Pat Roach:
In two of the participants who cried during the reunion scene, tears dropped first from the right eye and then [in] the third participant they dropped at the same moment from both eyes.
[11:30/38:24] Pat Roach:
During the “loss” scene tears started falling down first from the left eye in six of the participants and the right eye in four of the participants.
[11:40/38:24] Elan Dubrovski:
Seems pretty close to even.
[11:44/38:24] Pat Roach:
He goes on to say: “So I conclude there is no clear connection between which eye cries first and the specific antecedent. Hope this helps. Best regards, Ad."
[11:50/38:24] Pat Roach:
So while that’s not a very big sample size, I think the fact that Dr. Vingerhoets’ team has found exceptions to this, quote, “fact”, coupled with Dr. Nolfi’s opinion we can draw some fairly strong conclusions: This is a myth and [it] appears we here at TRC, with the kind help of these two doctors, are the first to really bust it.