Here are my thougths: After the first raise to $60, I would flat call his reraise to see the flop. If I had an overpair I would move in, with a King on the flop I would C-Bet but fold to any reraise. 

you should have folded to his raise preflop because you still would have had a good stack to work with whereas getting it all in here lost you the tournament

JJ is such a strong hand HU, I'm always ready get it alli-in. If you lost to brainless gambler, who was happy to stack off with 30% chance, it's not meaning you was wrong.  

Yaq are you leveling here?1) JJ is very strong, well ahead of almost anybody's min 3bet range, let alone the average unknown player.  It's fine to play for stacks regardless if you get min 5-bet, min 7-bet, etc.  Yes, he'll have QQ-AA sometimes, but for every one of those you see plenty 99, or K3, or AT, and so on.2) In higher stakes it's just as easy to stick it in and correct, as players often 3bet more balanced ranges that include fold equity.  Games are also more aggressive in the higher stakes, making JJ and easy hand to play for stacks preflop.  If you look at some high stakes players talk about mid to low PPs, you see guys like livb or skates in the past saying things like "I often 4bet shove 55 to a 3bet early on."  That's an indication of how aggressive the games can play.3) Don't worry about results, though in this example your results indicate that reraising with JJ was completely correct and wildly profitable (given that he held K3 you had around 70% equity which is a huge winrate).4) You don't flat call preflop vs the min 3bet with JJ because it is too strong and it is vulnerable.  You want value out of it now, since you often only have 2 cards to improve your hand, whereas your opponent will often have at least 6 (and possibly straight or flush draws to go along with it).  Overcard boards will be tougher to play, especially vs an unknown player.  In this specific hand, would you really fold JJ postflop with just Kx beating you on this flop and there being Tx, straight draws and diamond draws, most of which you're ahead of?  You'd either be prone to being bluffed out of the pot or beat by something as weak preflop as K3.  In short, get your value preflop with a hand this strong.  If you have something like 66, you can flat call and just play for your set odds/try to pick apart some weakness or get to showdown cheap, maybe even bluff catch on certain favorable boards.  But JJ is too strong to play that passively against an unknown, even against a 3bet.