Yes, it could be.
No, it can't happen

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Let's look at the black polygon's perimeter. In the beginning it's perimeter is $4$ .

If we color a cell which has $1$ adjacent black square, then the perimeter grows by $2$ .

If we color a cell which has $3$ adjacent black squares, then the perimeter decreases by $2$ .

So suppose yes, it could be. Then we have to color $63$ squares. Since by two coloring the perimeter grows by $4, 0$ or $-4$ , after the $62$ . coloring the perimeter will be divisible by $4$ , so after $63$ coloring, the perimeter won't be divisible by $4$ . (It will make $2$ remainder.) But the whole square's perimeter is $4\times 8=32$ , which is not possible.

So the answer is: No, it can't happen.