How does zinc affect fast neutrons?

Here's a plot of some cross sections for neutrons on zinc from the Evaluated Nuclear Data File at the National Nuclear Data Center. Blue is total cross section, $\rm Zn + n \to\text{anything}$. Green is elastic scattering (including scattering to excited states in the zinc nucleus, whence the structure), red is inelastic scattering, and grey is capture with gamma-ray emission. There are some modes missing from high-energy end, probably where the incident neutron knocks out another nucleon.

Here's a similar plot for hydrogen:

So for "fast" neutrons, 1--10 MeV, the cross section for interaction with zinc is a little higher than for hydrogen. However for all lower energies a hydrogen nucleus is more likely to interact with a passing neutron than a zinc nucleus by about an order of magnitude. (I'm doing some averaging-by-eye over the cross section structure in the keV-MeV energy range.)

Note that an important part of sustaining a reaction is slowing the neutrons down to thermal energies ($\rm 25\,meV = 25\times10^{-9}\,MeV$). For this process, zinc is not very effective, but not negligible either.

The usual place I go look up neutron cross sections is over at the National Nuclear Data Center in the Evaluated Nuclear Data File section. There you can get a variety of cross sections vs energy for Zinc or specific isotopes.

For Zinc overall, they have data from 10E-4 eV to 10 MeV, and there appears to be reasonable structure in the 10 keV to 1 MeV range, sith the cross section varying over several orders of magnitude in that range.