Assume wlog $m\geq n$. We have $\gcd(2^m - 2^n, 2^{m^2+mn+n^2}- 1) =$ $ \gcd(2^{m-n} - 1, 2^{m^2+mn+n^2}- 1) =$ $ 2^{\gcd(m-n, m^2+mn+n^2)}- 1$, by the well-known property $\gcd(2^{a} - 1, 2^{b}- 1) =$ $ 2^{\gcd(a, b)}- 1$. But $\gcd(m-n, m^2+mn+n^2) = \gcd(m-n, 3mn) \mid 3$, with $3$ possible, for example when $m=n=1$. Therefore the only possible values are $1$ and $7$.