It is possible that I'm doing the guy a disservice and he could just be young and stupid and embarrassed...
But he really sounds like an inconsiderate jerk. At best. I would recommend that you stop babysitting there, and definitely talk to a trusted adult, ideally your mother if she's fairly reasonable.
It's one thing to know, intellectually, that you have a right to say no to anything that makes you uncomfortable. But sometimes it isn't easy to know *how* to say no. That's where a little more specific advice from someone who knows you well could be helpful.
If, on the other hand, you think you do like him and want him to be your boyfriend, or at least someone you date, then you're going to have to talk to him about that. Maybe ask him why he doesn't even talk to you when other people are around.
But personally, I think he's bad news. And you deserve better. Stop going there.
(And what other people have mentioned, wondering why his mother is hiring you to babysit when he is available - That's a very good question.)