Pickled eggs do eventually go bad after about 4 months.
When you have pickled eggs they should be kept refrigerated once you open a jar of pickled eggs and refrigerate them until you use them all up.
It's best to eat the pickled eggs within a few months for the best taste and quality as overtime the pickled eggs lose the freshness and are not as good.
If the pickled eggs smell bad or taste bad then throw them out because if that happens they are unsafe to eat.
Homemade pickled eggs should always remain refrigerated because you could get botulism if they are not refrigerated.
Pickled eggs that are store bought and unopened can remain out in the pantry or just out of the refrigerator for up to 4 months and longer.
After about a year though the pickled eggs will start going bad so eat them up before then.