Feels like this is an extremely common situation, and more often than not the Muppet turns over Ax to beat your KK-QQ. I tend to just put him all-in no matter what the flop is if he's that short stacked, but that might be the wrong thing to do...
Obviously he's only going to fold if he missed completely (If the flop had been 3QK he would still have gone all-in, believe me) so maybe you should just check/call here. I mean either he's ahead, or drawing to 3-5 outs, so check/calling could have its merits...
Not to hijack but here's a good example of these sort of hands:
Seat: 1 parasense
Seat: 2 Stoneburg
Seat: 3 AlexIln
Seat: 4 Mattef
Seat: 5 Mr_Mitac
Seat: 6 sirmike
Seat: 7 frokenfraken
Seat: 8 naGon
Seat: 9 Maltow
Seat: 10 theofan (Dealer)
------------------
OpeningBetRound
------------------
parasense : Posts small blind 0.10
Stoneburg : Posts big blind 0.20
Mattef : Posts 0.20
naGon : Posts 0.20
AlexIln : Folds
Mattef : Checks
Mr_Mitac : Folds
sirmike : Calls for 0.20
frokenfraken : Calls for 0.20
naGon : Checks
Maltow : Folds
theofan : Calls for 0.20
parasense : Calls for 0.10
Stoneburg : Bets 2.00
Mattef : Folds
sirmike : Folds
frokenfraken : Folds
naGon : Folds
theofan : Calls for 2.00
parasense : Folds
------------------
Flop $5.40 [6c]
------------------
Stoneburg : Bets 3.60
theofan : Went all-in 1.65
Stoneburg : Receives returned betting money 1.95
------------------
Turn [4h]
------------------
River [7c]
------------------
Showdown
------------------
Stoneburg [ AC, AD, QC, 8D, 7C - One Pair ]
theofan [ 8D, 7C, 6C, 5H, 4H - Straight ]
theofan Showed Cards (Winning) 8.40
Stoneburg
theofan
I'm not sure what the best play is, but unless there's an obvious draw on the board maybe it's better to just check it (the exception being if you have an overpair of course).