AA are pretty good at checking luggage through to your final destination, even if on separate locators, so long as the gap at any one airport is 24 hours or less. So yes you should be able to check it through (on day 1) all the way to SJU. However what would happen if you made the connection, but your bag did not?

If handluggage or with checked luggage yes you generally stay airside and there's no futher security check so long as you stay in the same terminal building (plus a few exceptions). On these sectors you should only be searched at ORD and then SJU. SJU is relatively painless too.