There were some guys who did this. There was a pretty big disincentive to try it, of course, since the stigma (circa 1967 - 1972) of homosexuality would have been a pretty big obstacle, especially since even the rumor could follow a guy around and destroy his social life and any hopes for a career in his home town pretty effectively.
Mitchner had one of his characters in The Drifters consider that option, but I have no idea how close to accurate his fictional character behaved.
IF the local board had an easiy enough time finding bodies (and was sufficiently homophobic), a simple declaration might be enough to get an exemption/deferral. With a more hard-nosed or suspicious board, one might have to provide evidence in the form of a (presumably staged) homosexual act.
While I’m sure that it happened, I am equally sure that it was probably a really rare event. It is similar to going into a seminary to get a divinity exemption. I only knew one guy who tried that, and he couldn’t hack the seminary life well enough to stay in to keep his deferment. Given the number of guys I knew who did not get drafted (and the sometimes wild tales I heard of what they tried), I will note that the only places I heard about this were in “reports” in the paper or in fiction.
I suspect that most false deferments were accomplished by playing games to raise one’s blood pressure or talking one’s doctor into playing up a minor medical condition into something that could earn a IV-F.
Any comment on what “might” happen, today, would be pure speculation–especially since the military opposes re-instating the draft. (There have been charges that some of the people dismissed from the military for homosexuality in the last six months have been people who “came out” in order to avoid combat or overseas duty, however, that is also speculation.)