raising w 2nd best hand

[mumpsman]

I've been consciously trying to be more aware of opportunities to thin the field with a large pot and 2nd best hand. In a very loose game I'm utg with JJ. I raised, mid position reraises and the button caps. both blinds fold. Flop comes A 10 3 rainbow. all check to the button who bets. Is this a situation where I should be attempting to thin the field with a check raise? I've got to believe that the button has either A-big or KK or QQ. I'd have only 2 outs with my JJ.... thoughts?

What if I had AJ and the flop was K J 3 with the same betting (not that I'd automatically play AJ utg but just for sake of argument)

Thanks