Check what this says, especially about skunks and possums. See what it says about what the egg shells look like. From your description of the carcass I’d lean more toward possum, but you are looking at them, I’m not. The egg shells might be a big clue. Look for paw prints and scat too. Those are great clues as to what it is.
http://icwdm.org/inspection/livestock.asp
Personally I use peanut butter as a bait, just wad up some in a paper towel and toss it in the back of the trap. The main reason for his is that I always have peanut butter available. Other things work but I might not have them available. I’ve caught raccoons, possums, rabbits, and even a feral cat with peanut butter, never a skunk though, so I can’t recommend peanut butter if you think it is a skunk.
One problem I have is that critters often turn the trap over or work the bait out through the back or sides. I wired the trap to a piece of plywood on the bottom to give it more stability and keep them from coming in below it. Then I built a box top go over the trap so the only way in is through the trap door up front. If I ever trap a skunk I hope that box top will give me a chance to get it to my rain barrel without me getting sprayed.
I’d also look at how it is getting in and work on that. Just because you trap one doesn’t mean they are safe. It will remove one that is hunting your area so that is good, but I’ve caught critters on successive nights. That means more than one was hunting my area. And eventually another one will show up, say when Mama weans a litter and they go looking for their own territory.
Good luck with it.
