OP,
No.
It is a tactless move, and would likely be received as "hey, I know I haven't really accomplished anything of landmark mention in my life, but I'm pompous enough to feel that I should be distinguished from the other members of the pre-med community, and so here is a card for you to remember me by." Honestly, if I were a medical school dean, I would probably hold onto that card and wait for you to eventually apply to my program. If the committee were to invite you for an interview, I'd make sure I was the person to interview you, and after your interview I'd close with the following question
"One last thing before we end our discussion (pulls out 'pre-med business card'), this made its way to me some time ago, to circumstances in which I have since forgotten. Just out of curiosity, what was the purpose of making these cards and giving one to me?"
If the interviewee did anything other than laugh at themselves for it, I would automatically reject them. Seriously, to think that they could deliver some kind of meaningful reason for having done that, is just laughable, and as a potential entering medical student I would expect them to be able to own up to when they did something stupid. Trying to cover it up or play it off just wouldn't sit well with me. It's very, very, tactless.