[Doc]

I'd have raised pre-flop, which would give you some information at least. You're first into the pot with a strong hand, so if someone calls you, you've got something.

The turn is a brutal card for you. I think I'd overbet the pot here and hope he goes away. If he calls, I'm going to check it through. If he raises I'm out. Either way, no more money in the pot. Having raised pre-flop I'm not too scared of a full house, except for AJ, TT or AA.

As for your specific decision here, given the way you've played it:
The call wouldn't bother me too much here, although I think the re-raise to put him all-in is preferable. Either way you are committing to the pot. My gut reaction would be that I don't think I'd fold.

If we look at his possible holdings:

KJ, QJ - he is drawing to 2 outs from 44 cards - i.e. you are 22-1 fav.
AJ - you're dead
JT - ditto
TT - ditto
JJ - ditto
AA - ditto
KQ - split pot

You're basically being asked to commit $40 to win $64.
There are 6 ways he can have KJ, 6 he can have QJ, 8 he can have JT or AJ, 3 ways each for AA and TT, 1 for JJ and 9 he can have KQ.
In raw numbers that is 12 hands where you're a massive fav
23 where you're dead
9 where you split

You can probably slightly discount KJ (its not a great hand) and AJ (more likely to have raised by now), and likewise AA and TT

I'd say it is something like 9, 16, 9 before you account for the way he has bet the turn.

If we add up the cases, then 9xAx64 - 16*B*40 + 9*C*12

That gives 572A - 640B +108C, where A, B and C are the likelihoods based on his turn betting. If we assumed, for simplicity, that A and C are equally likely, then it becomes a +EV call to stay involved if there is a less than 45% (approx) chance that he has the house.

Its a very tough decision, and would come down to preference and your read of the other guy I'd say. Having considered it, I think you probably did the right thing though.