Unfortunately, I live in a subdivision, not the country....but with a very high wooden privacy fence and the back yard (the intended “killing ground”) sits in a sort of “bowl”, if you will; VERY private....but no way to use anything beyond an air rifle or slingshot.
I prefer the type you hand pump and more pumps = more power - keeps cartridges from leaking away if you only shoot occassionaly. Understand that one with enough clout to take out a squirrel will also "bark" pretty sharply so neighbors will be aware that something is afoot.
I know someone who lives in a similar setting. He picks off his squirrels from a bedroom window using a .22 rimfire loaded with primer-only ammo.
This ammo is very quiet (because there is no powder charge in the casing), travels at around 500 fps and is squirrel-deadly out to about 25 yards.
I forget the technical name for this ammo (Freeper help here!!) but you can get it anywhere, especially Cheaper than Dirt.
I’m in the city also....and actually a plain BB will kill squirrels (with a double or triple pump); my Remington has the capability for both BBs and .177 pellets, but I’ve made my kills with only the BBs.