I can't defend the appalling behavior of your street-corner preacher, for "all have sinned, and come short of the glory of God" and "If we say that we have no sin, we deceive ourselves, and the truth is not in us."
But C. S. Lewis had a good answer to your question about the square circle (or the more often heard question of "could God make a rock so large that he could not move it.")
"His Omnipotence means power to do all that is intrinsically possible, not to do the intrinsically impossible. You may attribute miracles to Him, but not nonsense. There is no limit to His power.
... meaningless combinations of words do not suddenly acquire meaning simply because we prefix to them the two other words, 'God can.'
It remains true that all things are possible with God: the intrinsic impossibilities are not things but nonentities. It is no more possible for God than for the weakest of His creatures to carry out both of two mutually exclusive alternatives; not because His power meets an obstacle, but because nonsense remains nonsense even when we talk it about God."