Is it legal to kill a grey squirrel?

Answer:

The Eastern Grey Squirrel and Eastern Fox Squirrel (the two most common species of grey squirrels in Idaho are listed as Unprotected nongame animals under Idaho code.  As such, under Idaho code 13.01.06 300.03 they: may be taken in any amount, at any time, and in any manner not prohibited by state or federal law, by holders of the appropriate valid Idaho hunting, trapping or combination hunting and fishing licenses, provided such taking is not in violation of state, county, or city laws, ordinances or regulations.

So in short, yes provided you have a valid hunting license and are not discharging a firearm in violation of city code.