Shoot whatever you have. There is no such thing as the perfect predator gun. I've killed coyotes with everything from a .17 to a .300wm and probably 2 dozen calibers in between. That said, I kill more predators with a shotgun than anything.
The important thing is knowing your situation. The area where I live in S Texas has a lot of brush (I guess that's why we're referred to as the "brush country") and coyotes don't like coming out into wide open spaces. Why would they? Their best hunting and best protection is in that brush. On the other hand, If I was headed to a friend's place in west Texas or Arizona, where there's lots of open ground and places to set up an overwatch, I need something setup for longer shots. However, I don't know any serious predator hunter that doesn't carry a shotgun to every set for those sneaky dogs that get right on top of you. Its very hard to get a LR rifle or a heavy AR up and twist around to get off shots on a coyote hauling ass away; but a good 12ga with a super tight choke and a 3" Dead Coyote load will easily get you out to 70yds every time. No matter where I'm calling, that gun is always in my lap or laying next to me, ready to be used on a second's notice. Calling coyotes is a part of my living and aside from my Winchester SX3 12ga, I don't have a dedicated coyote gun - I have several rifles I normally choose from:
AR15 SBR in .223
CZ 527.204 Ruger
Custom 6.5 CM
Ruger .17Hornet
The .17 Hornet is great on fur and is perfect for bobcats but I hate it for coyotes because it doesn't have the stopping power on marginal shots, which do occur. The .204 is terrific but mine just doesn't cycle as smoothly as I'd like so it's not my favorite rifle. My 6.5 is overkill and makes a mess of fur besides being unwieldy; BUT I can head shoot a coyote at 400yds with it and I DO love incredible accuracy. It's my go-to rifle for open country. My SBR is very handy and is my #2 gun, after my 12ga. I wish I could get better accuracy out of that 10.5" Noveske barrel but it only shoots about 1.25moa which makes it my least accurate rifle. I'm in the process of adding a micro red dot in an offset mount to help me with close range shots. I should have done this years ago, when the micros first came out. I think its almost a prerequisite if a predator hunter is only going to carry a single gun. I'm considering swapping to a 12.5" barrel to see if I can get a little better accuracy. However, I'm also considering building a upper in .224 Valkyrie and another in 6.5 Grendel to test out.
As with anything rifle-related, your sight(s) are probably more important than your caliber unless you're selling your fur, which I am not. Folks run down the rabbit hole on calibers but really, for most situations, the caliber is almost irrelevant. What matters is your ability to 'get on' an animal. Heck, even your trigger is often more important than the caliber you're shooting.
The last thing I'll say is that I thing the LR shooting craze (of which I am a fellow victim) has gotten both new and experienced predator hunters to jump into a often unnecessary rabbit hole. There's always talk of long range shots on coyotes and I'm not debating that it occurs but I will point out that these shots make up a tiny percentage of most shot opportunities. We all like the bragging rights to a coyote we dropped at distance; however, even in places where we can see coyotes at great distances, most experienced predator hunters will not start popping of at coyotes at great distances. Not once we have a shot or two under our belt to brag about. Why would we? There's no advantage to shooting a coyote at 500 yards. If you wait for him to come in as close as you can you will not only increase your odds of success, you stand a better chance of picking uo additional coyotes coming in. What's more impressive: a picture of a dead coyote you say you shot at 528yds or a bonafide triple? Would you rather end your day with 2 dead coyotes from LR shots or a giant pile of fur? I'm way more impressed with a pile of fur, and so are the landowners. I would always prefer the gun(s) that allow me to be the most effective at normal ranges.
Sorry for the very long post.