Well, if all I had was a Ruger 10/22 I would go and kill some coyotes. Buy a few boxes of 22 shells and practice up, first. I have killed several coyotes with a Ruger .22, they were close range though.
I shot three at one time, once, running together, and I got all three. The farthest one was probably only 30 yards and I did have a 4 power Redfield on it. Try to make a good first shot, headshot if you can, then pump 2 or three into the lungs, if you can. A scope would be good, but you can do it with iron sights.
Forget the 410 for reasons already stated. Better yet would be to borrow, one of your uncle's guns, even a .22mag would be an improvement. But if not, go for it. You might lose one, but the result will be the same, one less coyote. But if you might lose some sleep over a wounded coyote, don't do it, because there is definately a risk of having one run off, but then there always is, no matter what you are shooting if you mess up.