Uh yes. He is creepy. It does not matter if it is public knowledge or not, he should not be asking you. It is none of his business. If he REALLY wants the info, he will get it but at least make him work for it.
Next time, tell him just that. I am a nurse. I am here to treat you. If you need other information you can get it later, but I am here to do ................... whatever it is he are there for that time.
If he persists ask if he is refusing medical treatment. Because if he will not focus on his medical reason for being there, sounsd like a refusal to me.
But then again, I am not a nurse. Nor do I work where you do. MIght be best if you talked to your supervisor. He is most likely feeling you out to find out how far he can gow ith you and if he can compromise you. Don't take no crap to begin with and you won't get near as much later on!