By way of a more targeted hint there is this MSE post which shows how to solve the general problem. You need to compute the cycle index of the permutation group of the grid acting on the squares.
Here is the cycle index for $n=1$: $$ a_{{1}}$$ and for $n=2$ $$ 1/8\,{a_{{1}}}^{4}+3/8\,{a_{{2}}}^{2}+1/4\,{a_{{1}}}^{2}a_{{2}}+1/4\,a_{{4}}$$ for $n=3$ $$1/8\,{a_{{1}}}^{9}+1/2\,{a_{{1}}}^{3}{a_{{2}}}^{3}+1/8\,a_{{1}}{a_{{2}}}^{4}+ 1/4\,a_{{1}}{a_{{4}}}^{2} $$ for $n=4$ $$1/8\,{a_{{1}}}^{16}+3/8\,{a_{{2}}}^{8}+1/4\,{a_{{1}}}^{4}{a_{{2}}}^{6}+1/4\,{ a_{{4}}}^{4} $$ for $n=5$ $$1/8\,{a_{{1}}}^{25}+1/2\,{a_{{1}}}^{5}{a_{{2}}}^{10}+1/8\,a_{{1}}{a_{{2}}}^{ 12}+1/4\,a_{{1}}{a_{{4}}}^{6} $$ and for $n=6$ $$1/8\,{a_{{1}}}^{36}+3/8\,{a_{{2}}}^{18}+1/4\,{a_{{1}}}^{6}{a_{{2}}}^{15}+1/4 \,{a_{{4}}}^{9}. $$ Shading exactly two of the squares you want to compute $$[z^2] Z(G)(1+z)$$ which gives the sequence $$ 0, 2, 8, 21, 49, 93, 171, 278, 446, 660, \ldots$$ This is OEIS A014409 which seems to match your problem definition so this is probably correct.