Go with the 10/22....there are several aftermarket accessories for it when one day you decide you want a "match grade" rifle for shooting turtles.

I've got to agree with the .22LR vote--first of all, don't forget to check the used gun rack. I've bought some very good used .22's, all of them inexpensive and very accurate, often with decent scopes on them. One of my friends has a .17 and they are very loud and a missed shot will carry much further [ something to think about when shooting up into trees]. I've not owned a semi-auto so can't comment on them, but every .22 bolt action I've owned, Mossberg, Winchester, CZ, Remington and the Savage/Anschutz I use now, has been a superb squirrel rifle--the Seventeens are impressive, but I don't need that much noise, speed and expense to hunt squirrels----

Ruger 10/22. Got three of them, they all shoot great, accurate, reliable, and just fun to plink with as well.
-Jake

IMO the first thing you have to do is pick your cartridge and ammo. Personally I would go with a 22 LR, and I would choose sub-sonic rounds. These rounds are more than powerful enough to kill or to bark squirrels. A 17 HMR or 22 magnum is irresponsible in the squirrel woods. Way too much power, and way too much down range performance. Choose the slowest sub-sonic that shoots well in your 22.

When shooting sub-sonic's alot of them will not function a semi-auto rifle. So I do not believe a semi is a good choice.

That leaves us with lever actions and bolt actions. If you really like the faster action then go with a 9422 or a Marlin lever. That will push your budget. But they are the best. You can get a Henry in your price range, and alot of people like them. But personally the receivers and the plastic turn me off.

The next choice is a good accurate bolt action. I would look for a good used Remington bolt action 22 like a Remington 581. Or some of the other bolt actions that Remington made.

I would skip the Savage's and the Marlin's, but that is just my opinion. You have enough money to buy a really nice rifle so find that good used Remington 581 and you will have a fine rifle for life.

Then a scope. A simple scope is all that is need. I shoot alot of squirrels with open sighted rifles.

A 2X7 is more than enough power. But personally I would spend my money on a higher quality fixed 4X scope.

I see guys all the time with 3 or 4 or 5 or 6 different 22 LR's. I never understod that. I would spend my money on one top quality rifle before I would own that many different low end quality rifles. Just an opinion. Tom.

JMHO ,But that is the best answer to the OP question yet.