Heavy .22 cal pellets (18-25 gr) traveling close to 1000 FPS will easily dispatch squirrels humanely with either head or heart/lung placement. The difference between spring/nitro piston models and pre-charged pneumatics (PCP) are night and day. The double recoil of the former versions are noted to be strong enough to require scopes to be specially designed to take the bi-directional impulse. PCPs shoot like .22LR rifles with even less recoil. You can also get magazine-fed PCPs that only require re-cocking of a bolt for follow-on shots, and there are even semi-autos. High-end PCPs can go for $2-3,000, and are beautiful machines. These commonly store more than 4,000 PSI so electric pumps are almost required. Airgun stores and other sources can also supply scuba bottle refills at nominal fees. The other end of the PCP spectrum can pretty easily be filled to their 3,000 PSI storage level using a high-pressure hand pump. Yes, it will take you 100 strokes of what looks like a heavy-duty bicycle pump to fill the tank, but that will supply you with 20 to 50 shots before you need to repump again. Do check out the Pyramid Air site as they have bundles of lower-end PCPs and pumps right around the price point you've set. there are other reliable vendors on-line too (Airguns of Arizona, Utah Air Guns, etc.).
Keep in mind that some manufacturers and some vendors try to push supersonic velocities as an advantage. Shooting really light pellets will allow supersonic velocities, but the diablo shape of airgun pellets is inherently unstable at transsonic speeds. Supersonic pellets also issue the noisy crack associated with supersonic bullets, so they are much louder than subsonic pellets (and much less neighbor friendly). You will have better precision and results with heavy pellets just under the speed of sound. Good quality PCPs with good pellets are as accurate as .22LR out to ~100 yds. They are more precise than .22LR under 50 yds.
