Hello sausage646,
That's pretty sick. But I just don't see hands what he would Check/Raise like this. He never does it with 2pairs. So comes up the question either he does it with trips? Some of the trips ranges like A4 already beats us. And other trips I guess he would be inclined to raise us on flop or turn.
KK/AA ain't that likely. So basically QQ/A4 beats us, Q4/K4 ain't that likely unless he is very loose opponent. So practically it's up to some 54 type of hand since I don't see many other hands having the 4x. Has to be very loose.
So for conclusion it's hard to imagine him playing this way with QQ while KK/AA he would likely 3bet. But for the reason that your bet sizes were so small then we can likely even sometimes add QQ as well into his range.
So if we add some 54 type of hands into his range as well. We need ~28% equity for break-even:
Board: 4
K
4
A
Q
Equity Win Tie
UTG 71.43% 71.43% 0.00% { QQ, 44, A4s, 54s }
UTG+1 28.57% 28.57% 0.00% { JcTc }
So basically it's pretty much break-even Call unless we can put him on a wider range like 54o as well.
Best regards.