For varmints/coyotes/wolves, a .223 would be hard to beat. I've got a Browning with a 1:9" twist, and I have found that to be about ideal. I handload, and I can shoot 40 gr. V-max for bobcats, coons, and other small critters, 50 grainers for coyotes, along 55 gr. Sierra Gamekings and 60 gr. Nosler partitions for larger critters, such as hogs and deer.

For rabbits and squirrels, if you are planning on cleaning and eating them, the .223 is probably a little much. However, if you handload, you can load reduced loads using Blue Dot and make it shoot more like a .22 hornet or .22 mag. You might also use FMJ bullets on rabbits. If you get a good head shot in on them, it shouldn't damage the meat much.

I wouldn't use the .223 on squirrels, unless you are shooting them on the ground. The .223 bullets will travel a long way, even if you hit the squirrel. You'd be better off with a .22lr or shotgun for shooting up into trees.