Hero fold with Pockets Kings on the River to all-in overbet?

Here I had KK and my opponent had QQ.  I raised preflop and they reraised.  Our stacks were 300 big blinds deep and I thought I should keep my opponents range wider so I called rather than allin and risk my stack against Aces.  The flop came with a Queen and rainbow rag rag.  I checked the flop and he checked back.  I bet turn and was called after deliberation.  My first thought was I think they have a set of Queens.  On the river a blank came and I thought maybe they have 10s JJ and Im ahead or I could be behind AA also.  I bet anyway and was raised all-in.  Very tricky hand to play which I ultimately made the right fold because villain tabled a set of queens for a full house.  Please help street by street. Thanks Gil