I don't know much about this topic in particular, so I don't have any real details about it, but I've seen it referenced in so many places that it's made me curious. The only source I can actually remember is Forrest Gump, where one of Forrest's superiors loses his legs and later becomes homeless.
Why did so many Vietnam War veterans become homeless? Does this happen frequently to veterans in general?
The high rate of Vietnam veteran homelessness at the time of Forest Gump's premier can be traced in part to its concurrent occurrence with Vietnam vets entering the male age group most at risk for homelessness.
A large study was done by in 1991, roughly at the same time that Forest Gump was being produced, that found Vietnam veterans did in fact make up a large proportion of the homeless veteran population. This was a study based on 10,500 veterans at 43 different locations. Vietnam vets represented 50% of the homeless vet population, compared with 29% of the general population (meaning 29% of non-homeless vets were Vietnam veterans).
But there are two important findings. First, the proportion of homeless veterans and non-homeless veterans exposed to combat was the same. Therefore we can say that combat roles did not increase the rate of future homelessness. Most importantly, they concluded that the high number of Vietnam veterans among homeless veterans was due to their being in the 30-44 year old demographic, which is the age group with the highest rate of homelessness. Not surprisingly, they identified substance abuses and mental health issues among these homeless veterans, with non-white veterans experiencing them at a higher rate.
Your broader question about veteran homelessness may be a better question for r/askSocialScience.
Rosenheck, et al. "Vietnam Era and Vietnam Combat Veterans Among the Homeless," American Journal of Public Health 81, n.5 (May 1991).
While I'm not sure if this kind of reply is allowed, this thread has very good answers pertaining to the public's perception of the Vietnam War both during and after its conclusion. It also leads well into this thread which discusses exactly what caused the vilification of those who participated in Vietnam.
Edit: Per automoderator's request, I'll put in a quick ping to /u/HClay77 and /u/Bernardito, though there are many other commentors in that thread that would be worth contacting for more questions, the comments by these users are especially insightful.
To piggyback on this question:
Was homelessness a problem for veterans of World War 2 in the US?