I would fold this pre-flop. More often then not, you're just bleeding chips. Also, you often catch attractive draws which puts you in situations like this. I only play JTs with lots of limpers in the early levels or with a big stack mid game. It's also great to push since it runs great against calling hands.

Since the big stack didn't bet the flop, I doubt he has a Q. I think the overbet on the turn is an attempt to pick up the pot and you might be ahead here. Even if you aren't, the outs justify pushing in.