Trust me, we've tried several kinds of traps. And while you think your chickens are secure, a fox will expose the flaws in your safe plan. They're amazing. I can have a dog out guarding the chickens ALL day. Nothing happens. The dog comes in for five minutes and one is gone. Or we're here and look away.Live traps (box traps) are useless for foxes. You'd have to catch one by hand and stuff him in there to get a fox in a box trap. Old time trappers, using steel leg hold traps, can get them regularly. A few years back an experienced trapper caught 28 in one season at the farm. All with steel traps. I believe as he trapped the resident ones, others moved into the prime territory and were caught. That speaks to the skill of the trapper that he could get that many, and to the size of the unseen fox population.
My plan is to never expose a bird I value to predators. It take a lot of work and money to build coop that way, but it's worth it. I would be happy to give advice to anyone about predator-proofing, just send (or post) a bunch of pics of your coop(s).
In one year we have lost exactly two roosters. Both great birds and excellent flock protectors so that tells me they were doing their job, but you can't let your vigilance down for one second.
Our neighbors report similar stories. We've figured out the foxes must be watching. Unfortunately they aren't stupid.
Every time we lose a bird, we plug that hole until they figure something else out.