In the case of $f_c(z) = z^2 + c$, this recent preprint shows that we can factor $g_n(c) \pmod 2$ into irreducibles of degree $n$ in the odd case and something similar works for $n$ even too. In particular, see Theorem 1.2 and Theorem 1.6.
Unfortunately, the techniques don't seem to generalize in a straightforward manner for arbitrary $f_c(z)$.