When I was younger (better eye sight and enjoyed hunting) I used a Ruger Mk1 5.5" bull barrel with the iron sights. Mine is like a short rifle. As my youngest brother said, until he saw me shooting that Ruger he never knew pistols could shoot like a rifle.
There are bound to be other pistols that shoot pretty nice, too, my whole family bought Ruger Mk1's (except that youngest brother, who was never a hunter like the rest of us).
You'll want to find a place where they are eating. Not traveling from the nest/hollow tree to where they eat, but the place the come to eat. They will find a nut and often stop (barely moving) to cut it open and eat it. Makes it easier when they are sitting still. My dad taught us not to jump up and go grab the just shot squirrel as there might be others close that would see the movement and be spooked. Sometimes he'd shoot three or four before getting up to pick them up and put them in the pockets of his field pants or pouch on his vest.
My younger brother put a red dot on his Mk1 but I don't know whether he got to use it much before he passed away. They make mounts for the Rugers (sure they do for other .22 pistols as well).
It's fun. Sneaking up close, if you go that route, is tough and exciting.
Try multiple brands/bullet types in your pistol to see what is reliable and accurate. Believe it or not, my old Ruger shot Remington Thunderbolts like it was a laser. These days they have a bad reputation. I sighted in one of those tiny little single shot bolt action .22's (meant for youngsters) a couple of weeks back and it shot some very nice groups at 25 yds. with the iron sights. Sometimes I'd get two bullets tearing a bigger hole in the paper vs. three distinct bullet holes for the three shot groups.
So, again, try different brands to see what your pistol shoots best.
