.177 heavily regulated pcp. One that gets more than 40 shots per fill.
This type of air rifle will be backyard friendly in most cases. This caliber can destroy any squirrel. Head shots definitely bring out the kills that happen the closest to the point at which they are shot. But body shots are fantastic in this caliber too. And it is great for crow as well as pigeon.
I realize this is about rifle choice, but I'll mention one thing after reading this response. I have killed a bunch of squirrels with air rifles, and I have noticed something I find interesting regarding shot placement. Obviously, a shot through the brain is instantly lethal, no question. But, that kind of head shot virtually always results in a lot of nerve reaction, and the squirrel may flop around for 20 feet or more. Alternatively, a well placed heart/lung area shot is effective, and I have found that the squirrels hit in that location almost never move after hit. I can't explain it, and I'm sure they are not exactly instantly dead, but they just don't move around. I've observed this too many times to think it is an anomaly. The drawback, you need to know the squirrel's anatomy, and make a well placed shot, as one hit a bit too far back will not be a clean kill.
More on point, regarding your rifle choice, I'm afraid you will be challenged to find a satisfying rifle in that price range. Since you will be dispatching animals, and the range may be longish, you need the best rifle you can afford. For me, that always means a good trigger, which probably means more money than in your present budget. If you can find an acceptable rifle among the choices offered, that will be great, but I think might be better off long term to build up your budget broaden your choices.