To elaborate, I'm not talking about something that would immediately disqualify him, like a medical factor (if I remember correctly, being flat-footed was enough to get disqualified).
Instead, what I mean is something like a personality trait. Maybe it turns out that the man is undisciplined, doesn't take orders well, or doesn't work well within a unit. If a man has a personality unfit for a soldier, would the American military just make do? Or would he get kicked out?
In a draft, it's not like you get handed a uniform. It means you report for selection. People are turned down for all sorts of reasons, even in a draft. Poor vision, diabetes, epilepsy, heart problems, hearing problems, mental illness, being too short, or not strong enough to keep up. There are written tests, medical tests, physical fitness tests, and interviews to determine if you're a viable candidate for the military. Even though standards are usually lowered somewhat during a time of war, there are still standards to be met, and people who don't meet them will not be able to serve.
Knowing that, there were a lot of people, particularly in Vietnam, who tried to avoid service by failing the tests. For example, the Ishihara test for Colour Blindness includes a plate that can be seen by people of any colour blindness, and was used during the draft to find out who was lying to avoid military service.
Let's talk WWII, not Vietnam. My sources are How to Get Along in the Army by "Sarge", The Officer's Manual by Moss, and checking w/my husband, WWII buff, former history teacher, and former enlisted/NCO/officer.
First, they try to keep the bad eggs out to begin with. A criminal record disqualifies you for service.
If you get inducted, boot camp is supposed to get you into the habits of obedience and working as a group. Problem people got punished, including by other guys in the barracks. A "blanket party" consists of a group of guys walking up in the middle of the night, throwing a blanket over their target, then beating on him until they thought he got the idea they weren't putting up with him.
With further training to come, the service has about a year to grow someone up. If they are just incorrigible (say, they hope this gets them out), they will wind up being courted for disobedience, insubordination, and the like. This usually means stockade time, and stockade sergeants know how to beat a man up without it showing much (From Here to Eternity gets that right).
Eventually, with or without prison time, he will get booted out with a Dishonorable Discharge. This closes all government-connected jobs and most other employers don't want a discipline problem. Even a General Discharge wasn't too great for your future.