How man counts per hour to expect when using a he3 moderated neutron detector?
According to the 1966 UNSCEAR report, you should be expecting around 1e-2 cm-2 s-1 at sea level. High elevations or deep locations may differ.
Edit: I didn't think that He3's scattering cross section led to any meaningful moderation, and that it only had a meaningful absorption cross section in the thermal range - so you may find that the detector efficiency for cosmic background isn't that high.
I'm more curious as why you would be doing that to begin with.
BTW Cats are decent neutron moderators.
You can put the HE3 probe underneath a basket full of cats and you get improved detection efficiency.
What controls did you use to ensure the cats are not actually a neutron source?