What firearm would you recommend for hunting these?

I would recommend a 20 gage. A 12 gage has more range, and under some cirstances can put more pellets into a smaller area, but has as a trade off more felt recoil and typically a heavier gun to carry around. A 20 gage can be purchased with either a smooth or rifled barrel. For the game species that you have mentioned, I would select a smooth barrel, which should also come with a variety of choke tubes. Most firearms dealers will show you how to change these out. Typically, for rabbits, squirells, and turkeys, a full choke is preferred, as this places the shot pattern in a smaller area. For upland birds, a modified or improved choke can ter the pattern a little more, which helps reduce the aiming error of less experienced hunters. 20 gage shells also come in a variety of shot, as is available nearly everywhere ammunition is sold.