Facing a check raise with JJ and no overcards on the flop

You should always play with 100bb stack, use auto top-up.
Villain raises 4bb from SB, which looks weird.. Is that his standard size? Looks scary.
I would say that JJ is almost always a 3bet here. By calling you don't get the max value out of him and you let him play with his weaker hands that are more disguised postflop and then you lose a lot of money when he hits Also if you just call JJ+ because you don't want him to fold worse hands, what hands do you end up 3betting? You should have value and bluff 3bet range here and JJ is a clear value 3bet against SB.
OTF he could xr draws,sets, higher PPs. I would call here and re-evaluate OTT.

JJ is a pretty standard 3b for value.
On flop the min raise can be strong hand, draw... you can call and see turn ?

Call. This puts more pressure on him than you might think in case he is bluffing and turn comes bad for him. A raise is easier to handle on his part. He then still has two outs and pretty good fold equity in a rather big pot if he choses to reraise/shove. (It is also one of my perpetual tirades about not raising with bluff catchers or for information. Sorry)