Let me see if I got this right?

Preflop the pot is $4.

On the flop, you raise $4, villain calls for $4, making pot $12.

On the turn, you raise $8 (making pot 20$) and villain raises to 16 (making the pot $28).

So you need to call $8 for $28 pot. That's giving you a 3.5 to 1 call. (you got 9 outs for the boat, and one out for the four of a kind).

I'll let a more seasoned player correct me, but I think I'd have to call the miniraise, even though you are probably beaten. If you don't get the out on the river, fold to a substantial raise.

On a side note, I don't understand the villain's all-in bet, unless he made his flush and thinks you made a straight.