As mentioned containing your birds is likely the first step and as also mentioned electric fence might be your next logical step unless you or someone you know have experience trapping... particularly coyotes.
As for trying to catch the coyote in a cage...in can be done, but most people with experience will tell you that it is not worth the effort or the money. Firstly, you'd need a very large trap and they go for over $150 for the cheaply built ones found at the farm stores.
Now it depends what part of the US you are in, as coyotes can very in size, with them being larger in the midwest and eastern part of the country, but I think it's fair to say that a coyote of any size is very likely going to ruin the cage trap after one catch.... and if it's in the cage for any length of time overnight ... there's a pretty good chance it'll bend the bars on the door and work it's way out... unless the trap is extremely well built ... which again the $150 plus farm store versions are not.
Now in the fairness of disclosure, while I have trapped a number of coyotes using other methods, I have never attempted to catch a coyote in a cage trap...so what I've related above about not being worth the time and effort comes from others who have actually tried using cage traps for coyotes.
For a BYC member that as recently been setting cage style traps for coyote, I'll tag
@cmom ...maybe they have something more specific to offer, based on actual experience.