The .223 will work fine for everything you mentioned except squirrels. Just stay away from FMJ, they are not and never were intended to be hunting bullets. And like Icetrey said, shoot hogs in the head. Head shots on hogs are easy, because they don't have the quickness or range of motion in their neck like a coyote or deer.

I've had very good luck with 55 gr. Sierra Gamekings, but the .223 has a lot of options for ammo and bullets.