we had problems with squirrels/rodents
getting in the attic at night and making noises.

They do more than make noises. They destroy things.

One of the inspectors that came by said that
we had fascia gaps and gaps between the gutterboard
and roofboard. He also said that these gaps were common
in houses built in Georgia.
My question is: Should these gaps be covered?
Or is there any safety reason (overheating?) that they should
be there in the first place?

Yes, they should be covered to keep the varmits out. No, there is no
safety reason to keep them uncovered.
The easiest way to cover them is to staple on a sturdy
chicken-wire-like mesh. It comes in rolls at a good width already.
You lift up the bottom shingle and staple away.