I completed preflop cos i wanted to see what the button did if the bb pushed. if he folds i'm pushing not completing.

should i push the turn? As played is the push ok on the river? button is passive and likes to see it to the river.

PokerStars No-Limit Hold'em Tourney, Big Blind is t200 (5 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

saw flop|saw showdown

UTG (t4210)
MP (t2470)
Button (t3360)
Hero (t2350)
BB (t1110)

Preflop: Hero is SB with K, 8.
2 folds, Button calls t200, Hero completes, BB checks.

Flop: (t600) Q, 5, 2 (3 players)
Hero checks, BB checks, Button checks.

Turn: (t600) K (3 players)
Hero bets t350, BB raises to t910, Button calls t910, Hero calls t560.

River: (t3330) 3 (3 players)
Hero bets t1240 (All-In), Button calls t1240.

Final Pot: t5810

Results in white below:
BB has 8s Ks (one pair, kings).
Button has As Ac (one pair, aces).
Hero has Kh 8c (one pair, kings).
Outcome: Button wins t5810.