My thinking seems standard to me:
Dealt jacks in middle position - UTG limps, so I re-raise 3x + limp (4bb), UTG calls.
Flop
Flop comes with one over-card. I c-bet from position because villain might be on a flush-draw. If the flop were rainbow I might check to keep the pot small.
Turn
Introduces a second flush draw. I check for pot-control (if I had checked the flop, I'd bet here if it weren't an ace or if it brought in the flush)
River:
The backdoor flush comes in, villain bets about 1/3 of pot. I call because I think that the villain is trying to represent that he has hit the backdoor flush. Which seems silly since I bet 1/2 the pot on the flop.
Question: is this is a snap-fold to the villain's river bet, or is it a call?