QJ doesnt beat him (board paired) i think you have to call on this river. what hands do you put him on that beat yours? only thing a reasonable opponent wins with would be AA QQ or JJ but i think his range has to at least include AQ, KQ, QJ. If he has something like A6s i think you have to pay him off here and make a note to 3 bet him a lot. I dont think AA would check/ call the turn and check/ raise the river so i think you definitely have to call in this spot with the pot odds you are getting.

If you are going to flat call KK preflop i think you have to raise this flop without a read that your opponent will spew off chips with something like KQ in this spot.