Eggs that have been previously refrigerated in the store should always be refrigerated until used up.
I have however left refrigerated eggs out on the counter before by accident for 4 days and they were still okay to eat but it's best to go ahead and refrigerate them especially to make them last longer.
Fresh eggs that came direct from your chickens or someone else's chickens and have never been refrigerated can safely last for up to 5 weeks without refrigeration.