Yes, the claims are approximately true.
From a 1976 study:
Two stages of Vietnam drug use are identified-a period of increasing marijuana use followed by the 1970 influx of highly potent heroin to which 1/5 of the enlisted troops were addicted at some time during their tour. ... Since 95% of those who were addicted to narcotics in Vietnam have not become readdicted, the situation does not appear to be as severe as originally supposed.
Thus, 20% of US soldiers were addicted to heroin at some point during their tour and 95% of those addicted did not become re-addicted after the war and stopped using heroin.
This 2010 paper more specifically mentions only 5 percent of Vietnam War veterans who were addicted to heroin took heroin 1 year after the war. [See end of answer.] The authors also write (emphasis added):
Eighty-five percent of the men told us that they had been offered heroin when they were there—often quite soon after their arrival...Thirty-five percent of Army enlisted men actually tried heroin while in Vietnam, and 19% became addicted to it.
On 15 May 1971 (when the Vietnam War was ongoing, the New York Times published an article using the term "heroin addiction epidemic" chronicling the high use of heroin (emphasis added; paragraphs not all continuous).
So serious is the problem considered that Ambassador Ellsworth Bunker and Gen. Creighton W. Abrams, the military commander, recently met with President Nguyen Van Thieu on measures to be taken by the Saigon Government, including agreement on a special task force that will now report directly to Mr. Thieu.
John Ingersoll, the Director of the Bureau of Narcotics and Dangerous Drugs, also conferred with Mr. Thieu and other officials and returned to Washington, reportedly alarmed at the ease with which heroin circulates and fearful of the danger to American society when the addicted return craving a drug that costs many times more in the United States than it does here.
The epidemic is seen by many here as the Army's last great tragedy in Vietnam.
Some officers working in the drug‐suppression field, however, say that their estimates [for addiction to heroin] go as high as 25 per cent, or more than 60,000 enlisted men, most of whom are draftees. They say that some field surveys have reported units with more than 50 per cent of the men on heroin.
In the present day, both CNN and NPR write that 15% of US soldiers in Vietnam were addicted to heroin in 1971. From the second source (paragraphs not continuous):
In May of 1971 two congressmen, Robert Steele from Connecticut and Morgan Murphy of Illinois, went to Vietnam for an official visit and returned with some extremely disturbing news: 15 percent of U.S. servicemen in Vietnam, they said, were actively addicted to heroin.
Soon a comprehensive system was set up so that every enlisted man was tested for heroin addiction before he was allowed to return home. And in this population, Robins did find high rates of addiction: Around 20 percent of the soldiers self-identified as addicts.
"I believe the number of people who actually relapsed to heroin use in the first year was about 5 percent," Jaffe said recently from his suburban Maryland home. In other words, 95 percent of the people who were addicted in Vietnam did not become re-addicted when they returned to the United States.
The Huffington Post also quotes both the 15% and 20%. It would appear the 15% statistic is the estimate of 2 congressman whereas the 20% statistic is based on more accurate research after the war.
While visiting the troops in Vietnam, the two congressmen discovered that over 15 percent of U.S. soldiers had developed an addiction to heroin. (Later research, which tested every American soldier in Vietnam for heroin addiction, would reveal that 40 percent of servicemen had tried heroin and nearly 20 percent were addicted.) The discovery shocked the American public and led to a flurry of activity in Washington, which included President Richard Nixon announcing the creation of a new office called The Special Action Office of Drug Abuse Prevention.
To conclude:
Did 20% of US soldiers in Vietnam use "loads of" heroin?
19% of enlisted US soldiers were addicted to heroin "at some time during their tour." 20% is approximately accurate.
Did 95% of the US soldiers who were using "loads of" heroin stop using heroin afterwards?
By 1 year after the war, 95% of the US soldiers addicted to heroin stopped using heroin.
Why was this the case? The environment was different.
From NPR:
It's important not to overstate this, because a variety of factors are probably at play. But one big theory about why the rates of heroin relapse were so low on return to the U.S. has to do with the fact that the soldiers, after being treated for their physical addiction in Vietnam, returned to a place radically different from the environment where their addiction took hold of them.
"I think that most people accept that the change in the environment, and the fact that the addiction occurred in this exotic environment, you know, makes it plausible that the addiction rate would be that much lower," Nixon appointee Jerome Jaffe says.
From the Huffington Post (emphasis added):
Here is what happened in Vietnam: Soldiers spent all day surrounded by a certain environment. They were inundated with the stress of war. They built friendships with fellow soldiers who were heroin users. The end result was that soldiers were surrounded by an environment that had multiple stimuli driving them toward heroin use. It's not hard to imagine how living in a war zone with other heroin users could drive you to try it yourself.
Once each soldier returned to the United States, however, they found themselves in a completely different environment. Not only that, they found themselves in an environment devoid of the stimuli that triggered their heroin use in the first place. Without the stress, the fellow heroin users, and the environmental factors to trigger their addiction, many soldiers found it easier to quit.
You can read more about the effect of the environment in this excellent article written by Robins (who led the original research time authorized by Nixon). Quoting one sentence:
In the first year after return, only 5% of [soldiers] who had been addicted [to heroin] in Vietnam were addicted in the US.