I was anguished when I saw his min-raise that he had 77. The call was out of frustration mostly. What do you guys do when you see this happen? Should have been a fold?

Comments on turn and river are appreciated too. I was so caught up in my card coming like that that I wasn't exactly thinking clearly.

Party Poker No-Limit Hold'em, $ BB (10 handed) from (Format: FlopTurnRiver)

MP2 ($71.22)
MP3 ($50)
CO ($174.32)
Hero ($50.50)
SB ($15.76)
BB ($15)
UTG ($31.80)
UTG+1 ($82.98)
UTG+2 ($25.20)
MP1 ($50.65)

Preflop: Hero is Button with K, J. MP3 posts a blind of $0.50. SB posts a blind of $0.25.
4 folds, MP2 calls $0.50, MP3 (poster) checks, 1 fold, Hero raises to $3, 2 folds, MP2 calls $2.50, MP3 calls $2.50.

Flop: ($9.75) J, K, 7 (3 players)
MP2 bets $0.5, MP3 calls $0.50, Hero raises to $7, MP2 raises to $13.5, MP3 folds, Hero calls $6.50.

Turn: ($37.25) K (2 players)
MP2 checks, Hero bets $10, MP2 raises to $20, Hero calls $10.

River: ($77.25) 9 (2 players)
MP2 bets $14, Hero calls $14 (All-In).

Final Pot: $105.25

Preflop I would have just called.
At the flop the call looks fine. He can also be on a flush or straight draw. If nothing like this hits I would bet hard on the turn.
Your turn bet looks weak. I would have bet 20$. After your 10$ bet, his raise your call is stupid, why not put him all-in directly?

But in total I do not see anything ugly

MHFlush

3 bet the turn AI.

AK and QK also make sense for villian to min-raise.