Party Poker 0.50/1 Hold'em (6 max, 5 handed)

Preflop: Hero is BB with , .
UTG calls, MP calls, Button calls, SB completes, Hero checks.

Flop: (5 SB) , , (5 players)
SB checks, Hero bets, UTG calls, MP calls, Button calls, SB raises, Hero 3-bets, UTG folds, MP calls, Button calls, SB caps, Hero calls, MP calls, Button calls.

Turn: (11 BB) (4 players)
SB checks, Hero checks, MP checks, Button checks.

River: (11 BB) (4 players)
SB bets, Hero raises, MP folds, Button folds, SB calls.

Final Pot: 15 BB

Results in white below:
SB has 6s 4s (flush, eight high).
Hero has 9s Ks (flush, king high).
Outcome: Hero wins 15 BB.

SB check-raised the flop despite not knowing who would bet. Luckily for him, I was the bettor so he could trap the field. I thought he either had two-pair or a set on the flop. But then he checks the turn? Either he had a draw or wanted to check-raise again. For some irrational reason I thought he was trying to be tricky and check-raise again. I should have bet that turn to not give free cards.

When he led out the river, I put him on the flush. But then I raised? I should have certainly gone for overcalls here. SB could have had the a-high flush and MP and Button were calling stations (as long as it wasn't two bets on the river) so they probably would have called.

Would you have folded to a re-raise on the river here, since he might have had the A-flush?

Never. This isn't a four-to-flush board so he wouldn't necessarily be 3-betting with the nut flush, the pot is huge (I'd be getting 17:1 on my call) and he could be overplaying many hands such as a set, two-pair, or lower flush. There's at least a 8% chance he'd be overplaying one of those hands in this situation. Also, losing flush over flush is incredibly rare (when there's only 3-flush on board), especially when I have the K-high flush.