We use a momentum-dependent optical model potential to analyze the annihilation cross sections of the antineutronn on C, Al, Fe, Cu, Ag, Sn, and Pb nuclei for projectile momenta p lab 500 MeV/c. We obtain a good description of annihilation cross section data of Barbina et al. [Nucl. Phys. A 612, 346 (1997)] and of Astrua et al. [Nucl. Phys. A 697, 209 (2002)] which exhibit an interesting dependence of the cross sections on p lab as well as on the target mass number A.We also obtain the neutron (n) non-elastic reaction cross sections for the same targets. Comparing the nA reaction cross sections σ nA rec to thenA annihilation cross sections σn A ann , we find that σn A ann is significantly larger than the σ nA rec , that is, the σn A ann /σ nA rec cross section ratio lies between the values of about 1.5 to 4.0 in the momentum region where comparison is possible. The dependence of then annihilation cross section on the projectile charge is also examined in comparison with the antiprotonp. Here we predict thepA annihilation cross section on the simplest assumption that bothpA andnA interactions have the same nuclear part of the optical potential but differs only in the electrostatic Coulomb interaction. Deviation from a such simple model extrapolation in measurements will provide new information on the difference betweennA andpA potentials. PACS numbers: 24.10.-i, 25.43.+t, 25.75.-q, 2