If it's in a trap, and you shot it with a shotgun, you're not going to have much trap left. If it's a metal trap, I'd worry about the pellets deflecting and ricocheting. And you'd have to be careful of what was behind the trap as well.
If you can get a clean head shot with a .22 short and the animal dies instantly, it won't spray. But if it doesn't die instantly, it absolutely will spray in its death throes. A skunk is a small animal with a tiny brain; getting that perfect shot is not a sure thing at all.
And you have to figure out how to get close enough to get the shot before the poor skunk panics and sprays.
Fortunately this is all still hypothetical for us, but I am very much preparing for the day when we find a skunk in the live trap.
On another note, I'm really interested to hear that a skunk was killing chickens, I had no idea they'd do that. Eating eggs doesn't surprise me, but killing chickens really does.