= Solution
Let $z=2d$ be the coordination number of the hypercubic lattice. The mean interaction energy per site is $(Jz/2)m_em_o=Jd,m_em_o$. Define
$$
I(m)=\frac{1+m}{2}\log\frac{1+m}{2}
+\frac{1-m}{2}\log\frac{1-m}{2}.
$$
The two sublattices each contain half the sites, so the mean-field free energy per site is
$$
\boxed{
f(\beta,m_e,m_o)
=Jd\,m_em_o-\frac{h+g}{2}m_e-\frac{h-g}{2}m_o
+\frac1{2\beta}\{I(m_e)+I(m_o)\}.}
$$
The entropy terms are the binary-spin mixing entropies and the positive $J$ favors opposite sublattice magnetizations.
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