Hydrogen is pretty good at moderating neutrons down to thermal energies (eV range, ie room temperature) via elastic scattering, but gasses don't really have enough density to do a very good job. If you really want to protect something from neutrons you just coat it with boron. A mm coating of the stuff will keep out pretty much any common source of neutrons.
It's very surprising to see how efficient boron is. I thought neutron shields (paraffin, water) are supposed to be very thick. Maybe boron does the job via a different mechanism?
Boron's probability to capture a neutron is astronomically high, that's why you can get away with so little. Environmental sources of neutrons are actually pretty rare normally and most neutrons you do see will be pretty low energy and won't have a huge amount of penetrative power. A thin layer of boron will pretty much stop them. Pyrex (like the stuff baking dishes are made of, which is borosilicate glass - glass with boron added) is actually commonly used as control material in nuclear reactors.
So, if we had more hydrogen (either free or compounds) in the air this would not be the case, right?
The column of air on top of your head is equivalent (in terms of mass) to a column of water 10 meters tall, with the same base section area. But the composition is quite different, of course - the only major component they have in common is oxygen.