You can do whatever the fuck you want to do since you are the one making the hiring decision.
As long as the reason for not hiring the applicant does not become public you will be safe.
If someone finds out that you discriminate based on military service your ass will be sued and you will lose everything you own. This is as it should be. I would be happy to fill out an application to help with this occurance.
Military service does not equate to having mental problems or being a problem employee. You need to get over that delusion right now. There are just as many problem employees who are. Oh, Lets use as an example here..... Southern, Yankee, White, Black, Native American, Surfers, Goths... Are you starting to get the picture that you have pushed one of my hot buttons?
Discrimination in any form is not acceptable. Judge the person based on his ABILITY not his history.
If I was smarter than you, I'd just stop here with the advice to to a Google Search for the phrase "veteran discrimination"