A 20 gauge is fine for squirrels and is widely used for turkeys...by muzzeloaders. I think most centerfire hunters use 12 gauge guns because they are so much more lethal, and the availability of modern 12 gauge guns isn't an issue. Muzzleloaders use .20 gauge guns because 20 gauge tubes are available from so many sources. A 20 gauge has and will harvest anything from squirrels to deer. If you want to shoot turkeys or limb rats out to 25 yards, a 20 gauge will do the job. If however, you want to extend your range for turkeys, or shoot flying, a 12 gauge is preferable.
A 12 gauge will throw ball as well as a 20 but with more felt recoil. A 12 gauge ball of say 715 diameter will weigh in the 500 grain neighborhood, whereas a 20 gauge ball of .600 diameter will weigh in around 300 grains. With 3 drams of powder behind a 500 grain ball you will feel more recoil than with 3 drams behind a 300 grain ball.
I think smoothbore roundball accuracy is problematical at any distance beyond 50 yards, and haven't noticed 20's shooting any better than 12's. If you're wife or children share your interest in muzzleloading, a 20 gauge may be somewhat more manageable for them, but frankly gun fit determines recoil as much as bore size. I have taught hundreds of boys and girls to shoot flint trap, and have always trained them on a 12 gauge. I always tell them they have to be twelve to shoot a 12 gauge and I've never lost a student because of recoil. I have them shoot 3 drams of FFG and 1 1/4 ounces of shot, and they break more clays than shoulders. Generally kids are result oriented, and they will like what they do well at, and dislike what they do poorly with. They prefer a flint 12 gauge to a 20 gauge double, because they break more birds with the 12.