$10 NLHE 6-max: Top two pair on river - check back or bet?

Villian Stats (VPIP/PFR/AF): 17/13/3

pokerstars Zoom No-Limit Hold'em, $0.10 BB (6 handed) - PokerStars Converter Tool from http://flopturnriver.com/

Hero (Button) ($24.06)
SB ($10)
BB ($36.62)
UTG ($12.23)
MP ($29.26)
CO ($10.28)

Preflop: Hero is Button with Js, Qh
3 folds, Hero raises to $0.20, 1 fold, BB raises to $0.70, Hero calls $0.50

Flop: ($1.45) Jc, Qc, 10c (2 players)
BB bets $1.03, Hero raises to $3, BB calls $1.97

Turn: ($7.45) 6h (2 players)
BB checks, Hero bets $4.50, BB calls $4.50

River: ($16.45) 7h (2 players)
BB checks, Hero ?

Interested to see what people would do here? Is it a straightforward check or is there any value in betting? I think there are a few worse hands in villain's range that he might call with here, maybe AA,KK AQ. But is it likely he has me beat, maybe holding AK, QQ, JJ, TT, and does not bet on the turn and river because he's worried I have flopped a flush?

Jam river. Draws missed, button vs. blind, villain will call down light.

Yuk we are pretty deep to shove 2 pair here. We still have over 150bb left behind here do we not.
The problem is if you bet anything small and he has the Ace of clubs he can bluff shove over you and you have to fold. Unfortunatly its a card easily in his range. Only because we are so deep and he looks aggressive Id check behind but I'm naturally cautious and I always seem to get burned on monotone flops.
Tbh I think fold pre looked good to me I hate QJ such a trap hand and 17/13 at 6max zoom is tighter than most.

Thanks for the responses guys. I decided to bet 1/2 the pot on the river and he called with AK (with no club)! I can't understand why he would not protect a flopped straight on a monotone flop. It makes no sense! If he's putting me on an unlikely flopped flush, why is he proceeding with his hand all the way to showdown? If he isn't, why does he not seek to protect his hand and get the money in whilst it's still probably the best hand?
I'm think what I'm trying to say is, did the villain play this hand in an unusual/bad way? Could I ever put him on AK (specifically AK without a flush draw)?

Looks likely your flop raise on the monotone board spooked him into just getting to showdown as cheaply as possible.
Ak explains the 3bet and backs up with his stats that this was a fold preflop.
If you had shoved on the end he might have folded considering the strength you showed on the flop and turn. It shows the influence of the monotone flop and how you can use it.
It's difficult to put him on AK as he might also have played the nut flush draw or even bottom set this way.