He seems to like his hand, he's a solid player, nothing stupid he's done. What would you do?

He could have a full house already or KQ, made flush, or even a split.

Do I push here?

PokerStars No-Limit Hold'em, $0.50 BB (9 handed) from (Format: Bet The Pot)

MP1 ($13.75)
MP2 ($51.60)
MP3 ($50.70)
CO ($68.95)
Button ($10.60)
SB ($49)
BB ($40.50)
UTG ($17.65)
Hero ($48.50)

Preflop: Hero is UTG+1 with , .
1 fold, Hero raises to $2, 5 folds, SB calls $1.75, BB calls $1.50.

Flop: ($6) , , (3 players)
SB checks, BB bets $3, Hero raises to $10, SB folds, BB calls $7.

Turn: ($26) (2 players)
BB bets $5.5 hero ??

On a monotone board, I probably just call on the flop and see if a non-club turn hits. Say he has a A[c]. They'll call your flop raise everytime so no fold equity there and you are building a pot you cannot control.

As played, I probably just call his turn bet.

Push!

He had : :[9c]

I hit an A on the river

I push that too.