The 22 magnum is not enough for coyotes. Yes, with a perfect head shot it'll drop'em in their tracks, and a perfect heart/lung shot will kill'em, but in most cases you'll have a long very hard tracking job, with the majority being lost.
I'd say go with the shotgun. I've dropped coyotes dead in their tracks up to 40 yards with my fox shotgun which is a lowly 870 Express Magnum 20 guage loaded up with Federal 3" magnum #2 shot(full choke). If my 20 guage can do that good, then a 12 guage loaded with 3" magnum #2 shot or bigger should give you at least a few more yards of dependable killing range.
I'm likely in the minority here, but I wouldn't trust #4 buck, or any other buck shot for coyotes. Unless you can reliably put several of those pellets in his chest area at a given range, a coyote can take a hard hit, and still run like he hasn't been touched. I know the heavier pellets carry more energy, but with a shotgun, it's a gamble as to where exactly in the coyote that energy will be delivered.
I'm a firm believer in the awsome knockdown power of a shotgun, but I believe you get the best effect by hitting the vitals, the head/neck, and preferably hurt the running gear some to boot. If you put 10-20 pellets of #2 shot in those areas, a coyote pretty much just flops over right there in my experience. You'll end up with more holes in the hide, but they'll be a lot smaller than buckshot holes. 20 guage 3" magnum #2 shot usually ends up stopping against the hid on the off side of a coyote at 30-40 yards in case you're wondering.