A jar of pickled eggs can remain out of the refrigerator as long as they are not opened.
If you buy already made pickled eggs that are sealed in the jar they do not require refrigeration until you open the jar.
Once you open the jar of pickled eggs they must be kept in the refrigerator when not eating them or serving them to prevent spoilage.
Homemade picked eggs should always be refrigerated because they can grow bacteria and make you sick.
You can get botulism from pickled eggs that are left out at room temperature longer than a few hours.
Pickled eggs that are commercially made and sealed can remain out at room temperature until you open the jar.
Once you open the jar of pickled eggs they must be refrigerated.