Preflop
3-bet is fine with this hand. Out of position sizing could be a little bigger like 55-60c.
Flop
C-bet is standard, sizing is fine.
Turn
When they both call, I think, its highly likely, at least one of them flopped an overpair, and I dont think, I can get that kind of hand to fold. So at this point I would slow down and check and basically hope, that they somehow check behind, or that they give me a price, I can call. Instead you bet very small, which is sometimes called a "blocker bet". I dont think, they are ever going to fold to this sizing, and I also think, an overpair will raise all day, so I really dont like this bet.
You do in fact get raised, and now you move all in. Here there seem to be some confusion about terminology, because you talk about implied odds. But implied odds is the money, you will win or lose on future betting rounds, so when you move all in, there are no implied odds.
The reason to move all-in is for fold
equity, and since the guy has committed most of his stack already, I would expect very little fold equity, even though this time he actually did fold. Which can only mean, that he either misclicked or timed out, or he was on an absolute stone
bluff, which makes no sense for him to have after overcalling the flop bet.
But anyways I would not jam here, because I think, the vast majority of the time you are just putting your last 30BB in behind. I would however call, because assuming, that BB fold, which I would expect him to do most of the time, you are paying around 2$ to win a pot of 8,5$, so you need less than 20% equity, and with a 12 out draw, this is more than good enough even just in direct odds. I also think, you can play the river very well by essentially folding, when you miss and getting paid, when you hit.
Results
This is just a really bizarre hand with the way, everyone played it. BTN apparently raising the turn without even having a decent draw, BB slowplaying top set, and you making some weird blocker bet on the turn. At the end of the day the flop was just a classic set-up though, and its completely standard to go broke with your hand, when you dont improve.