Got a skunk that is getting my chicken eggs by the house. How can I set a trap and dispatch it without it spraying?

That's hit or miss on the spraying. I hears the no spray traps work but don't know how you check to be sure what's in the trap before giving it a bath.

Might try a conibear no guarantee it won't spray but maybe less likely

Wrapped cage and a long bath.

Never had one spray in a wrapped cage. Only leak was a young one in an unwrapped cage.

Catch in a cage trap, cover with an old blanket or something on that order, take it where you need to.
What you do with it then is up to you.
They don't spray , but make sure you carry the blanket in front of you, covering up your profile.

Wrapped cage saves the need for a blanket.

One contractor bag will wrap twice, leave the back open, around a standard raccoon sized havajunk/safeguard/tomahawk. Secure it well with duck tape and it should survive four or five skunks before replacement is needed.

When moving the cage always approach from the side and talk softly to the skunk. Main idea, skunk doesn't see you as a threat. You can even release said skunk by straddling the cage while opening the door and using a bungee or clip to hold it open while you retreat. Skunks will be in no rush to exit.

When caging skunks positive forced sets are best so set your trail pinch points and don't be afraid to block them in or use some drift fencing.

Covered cage trap like warrior states. Slow and deliberate movements when transporting them and don't let them see you. If they can't see you they don't have anything to shoot at.

Conibear is no guarantee but best option I would think. At least the skunk will be dead. Probably will spray the area.

Shotgun.(JK)