I think it's a question that cannot logically be answered because it is based on a contradiction of logic, but I'll ramble on about nothing anyway:
Can a circle be a square at the same time it's a circle? Perhaps God could do it. He could certainly make one look like the other. But to make a circle that was actually square, however, seems a logical impossibility (and if there was a shape that one couldn't identify as being either a circle or a square, it would be called something else).
By definition, a square has four equal sides which meet at four right angles. A circle has no corners. A square circle is therefore self-contradictory (duh). God cannot theoretically make something with two contradictory natures. It's possible, however, that there could be another reality in the same space as ours, in which circles in our reality correspond to squares in the other one, but that's something entirely different.
Now, I guess I can't really provide an answer to your question, other than claiming that God cannot do the logically impossible, which cannot be proven thanks to our limited ability to reason.
As for the rock that God himself couldn't move: it's the same as asking why an omnipotent God couldn't deprive himself of his omnipotency (or change his intrinsic nature in any other way). The only answer is somewhat (read: completely) circular, and that is that he cannot do so because it is part of his intrinsic nature.
tl;dr: As above, beats me!