OUCH!!. Yes, this really is a rough spot. I definitely think that you played your QQ correctly preflop and on the flop. I would agree with you that he is unlikely to be holding AA or KK. From looking at the villain's stats, I can't imagine him just cold calling the preflop raise and taking a chance on going to the flop 4 or 5 handed with AA/KK. I also disagree with Delvuter about the villain possibly holding AJ. Once you 3bet, he is getting just over 1 to 1 pot odds
and that ain't enough to call with AJ, even if he has position. (This is even more true if he knows you to be a TAG). Given all of this, I think he has a fairly narrow range here. Perhaps AK, 66-QQ, and MAYBE AQs. Once we lead out on the flop with a good sized bet and get called, alarm bells would be going off in my head. I think the villain is too tight to float the flop with air so we should assume he has something he likes. Since there are no flush draws on the flop, I think there is a good possibility that we are up against a set. Obviously, if we check the turn, he is likely to bet no matter what he holds so we are basically going to be done with the hand. However, I do think check/fold is the best choice here.
P.S.... This is EXACTLY the way I would have played this hand if I were holding JJ......EXACTLY!!