No. The draft was not used as a means of punishment, but some judges used to allow accused or convicted prisoners to enlist in a branch of the armed services as a way to avoid punishment.
This policy has a long history. During the Spanish-American War, for example, an Atlanta judge released several moonshiners from jail on the condition that they enlist in the Army. The local revenue official said "they are especially adapted to guerrilla warfare." During World War II, one of the men who enlisted in the Army under this policy was a 28-year-old who impersonated an Army colonel.
The policy has been attempted as recently as 2006, but with limited success because of changes in Army policy.
According to Eugene Fidell, former president of the National Institute of Military Justice, some soldiers who enlisted in the military as part of an enlist-or-prison program then committed crimes. They then claimed as a defense under the Uniform Code of Military Justice that they weren't legal enlistees because they had been forced to join. Fidell has subsequently gained a measure of fame as the person assigned to defend Bowe Bergdahl.
Today, the Army must issue a waiver to allow someone who is facing criminal charges if they express a willingness to enlist.
There are likely others here better able to speak on the relationship between the US legal system and the US army in the Vietnam War era (the directory of flaired users suggests /u/thinmountainair - hi!), but I can provide some context to Springsteen's 'Born In The U.S.A.' (the song this lyric is from). Firstly, Springsteen seems to have written 'Born In The U.S.A.' in 1981-82, a couple of years before Born In The U.S.A. the album was released in 1984. According to Peter Ames Carlin's relatively clear-eyed biography Bruce, Springsteen randomly met Ron Kovic, author of the memoir Born On The Fourth Of July, in 1979. This led Springsteen to put on a fundraising concert for the Vietnam Veteran's Association in L.A. in August 1981, in which veterans, often with injuries, watched from a 'specially built deck on the side of the stage'.
Springsteen himself was not a veteran - Carlin portrays him as telling comic stories onstage through the 1970s of his draft dodging - and by the early 1980s was often writing storytelling lyrics from the perspective of narrators - often unreliable narrators. Presumably the lyrics of 'Born In The U.S.A.' were influenced by the big impression that Born On The Fourth Of July made on Springsteen and on Springsteen's experiences with actual, often incapacitated, veterans organising the August 1981 fundraising concert.
One other possible influence on this line is Jimi Hendrix. Biographies of Jimi Hendrix (Room Full Of Mirrors by Charles Cross, Crosstown Traffic by Charles Shaar Murray, Jimi Hendrix: A Brother's Story by Leon Hendrix, etc) usually tell the story that, in 1961, Hendrix was caught riding in a stolen car, and that the legal authorities gave Hendrix the choice of jail or the army. Given this choice, Hendrix picked the army (Jimi Hendrix is usually portrayed as being temperamentally unsuited to being in the army, and so biographies usually do need to explain why he joined in the first place). If this story is broadly accurate, presumably some 1960s judges were dropping charges on the condition that the accused joined the army.
Bruce Springsteen was a fan of Hendrix; the Carlin book talks about him playing in a Jimi Hendrix Experience/Cream style 'power trio' called Earth in 1969 who played Hendrix covers as well as originals. So it's not impossible that the story about Hendrix influenced that particular line. Alternatively, not all soldiers in Vietnam were conscripts; soldiers could also enlist of their own free will. Presumably some might have done so in order to escape various 'little hometown jams' of a personal rather than legal nature.