Notice that as soon as the first three rows are filled, there is only one remaining way to color the grid to satisfy the condition. Number of ways to select an even number of squares to color black in each row: $1$ way for zero squares, $6$ ways for two squares, and $1$ way for four squares, totaling $8$. So our answer is $8^3=512$.

Answer = 2^((4 – 1)^2) = 2^9 = 512