Can We Use Parity?

Consider an 8×88 \times 8 grid of squares. We color every square blue or green such that 3 of the 4 squares on the corners of the grid are blue and the fourth is green.

Will there always exist a 2×22 \times 2 square inside this grid such that there are an odd number of squares of each color inside?

Bonus: Generalize this for the n×nn \times n grid.

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