60 - (4+6+25) = $25 in your stack left on the river?
I think you miscounted the pot.
$4 preflop, 3 to the flop, no SB = $12.5
3 players call $6 = 12.5 + 18 = 30.5
Turn: 3 players call $25 = 75+30.5 = 105.5
River: you're getting 5:1 to call.
Questions:
Should the button have folded?
Hard to say -- BB could have 22/33, but more likely a flush. He was getting 2.4:1 on the turn to draw, but I'm not sure why he called -- it's possible that he somehow has some kind of straight draw to go along with his flush draw (A

). OR, he was counting on UTG coming along for the ride as well (3.2:1 + implied). Button put himself in a bad spot by not getting another $25 into the pot on the turn (unless i misunderstand what the stacks are).
Should the button have raised more or less at the turn?
I would probably raise a little more on the turn so that your river decision is easier. The pot is already $36 and your call will make it $42 -- you only have $44 left in your stack. Moving all-in for a raise of +44 isn't unreasonable if you suspect someone is on a draw. Raising to $25 puts you in a tough spot on the river -- you're getting 5:1 but you're not sure if it's a value bet (JJ/TT/flush/straight) or a bluff (or a weird slowplayed hand you beat like 22/33).
Should the button have raised the flop?
Not sure what this would have accomplished, unless you were planning on making a big bluff on the turn if you missed your 2-outs. If you raise to $18, you'll bloat the pot to like 48~66 (depending on # of callers) and only have $32 left in your stack -- if 2 people call, i'm not sure you can force out a flush draw on the turn (this is bad, cuz he'll probably have 2 overcards to your 44 as well).
Should the BB have bet more or less at the river?
If stacks are $60, then 60-(4+6+25) = 25, so he should bet that. If he has a flush, that's a perfectly good bet :P
Should the BB have raised the turn?
With a straight? Probably not (i.e. nobody else is probably on a flush draw cuz they are betting!). With a straight+flush draw? Probably not. With top set TT/32/33/22 -- moving all in is probably best.
What did everyone have?
UTG: Maniac? 2 cards -- anything from medium pocket pair to AK.
Button: 444 (obviously) -- but it's very well disguised.
BB: flush/22/33/JJ/TT/straight/crazy bluff/AA/KK/QQ