  We use a momentum-dependent optical model potential to analyze the
annihilation cross sections of antineutron $\bar n$ on C, Al, Fe, Cu, Ag, Sn,
and Pb nuclei for projectile momenta $p_{\rm lab}$ $\lesssim$ 500 MeV/$c$. We
obtain good description of annihilation cross sections data of Barbina {\it et
al.} [Nucl.~Phys.~A {\bf 612}, ~346~(1997)] and of Astrua {\it et al.}
[Nucl.~Phys.~A {\bf 697},~209~(2002)] which exhibit an interesting dependence
of the cross sections on the $p_{\rm lab}$ as well as on the target atomic mass
number $A$. We also obtain the neutron ($n$) non-elastic reaction cross
sections for the same targets. Contrasting the $nA$ reaction cross sections
$\sigma^{nA}_{\rm rec}$ to the $\bar nA$ annihilation cross sections
$\sigma^{\bar nA}_{\rm ann}$, we find the $\sigma^{\bar nA}_{\rm ann}$ is
significantly larger than the $\sigma^{nA}_{\rm rec}$, that is, the
$\sigma^{\bar nA}_{\rm ann}$/$\sigma^{nA}_{\rm rec}$ cross section ratio lies
between the values of order 1.8 and 3.8 in the momentum region where comparison
is possible. The dependence of the annihilation cross section on the projectile
charge is also examined in comparison with antiproton $\bar p$. Here we predict
the $\bar pA$ annihilation cross section on the simplest assumption that both
$\bar pA$ and $\bar nA$ interactions have the same nuclear part of the optical
model potential but differs only on the electrostatic Coulomb interaction.
Deviation from such simple model extrapolation in measurements will provide new
information on the difference between $\bar nA$ and $\bar pA$ potentials.
