It depends on how you define metric on the product space. Say, if d_{X x Y}((x_1,y_1),(x_2,y_2)) = d_X(x_1,x_2) + d_{Y}(y_1, y_2), then h(AxB,CxD) = h(A,C) + h(B,D); if d_{X x Y}((x_1,y_1),(x_2,y_2)) = sqrt{d_X(x_1,x_2)^2 + d_{Y}(y_1, y_2)^2}, then h(AxB,CxD) = sqrt{h(A,C)^2 + h(B,D)^2}; if d_{X x Y}((x_1,y_1),(x_2,y_2)) = max(d_X(x_1,x_2), d_{Y}(y_1, y_2)), then h(AxB,CxD) = max(h(A,C), h(B,D))

See attachment.

But I am not sure at all that this would for the first two examples I wrote.