Once you pickle eggs the pickled eggs will keep and be safe to eat for up to 4 months as long as you keep the jar of the pickled eggs sealed properly.
If you want to keep the pickled eggs longer than 4 months you should refrigerate them and they will last for 6 months to sometimes a year.
Pickled eggs taste better when refrigerated too so I always just keep my pickled eggs in the refrigerator because it helps them keep their flavor and makes them last longer.
Sometimes I'll keep a large jar of pickled eggs for 7 months while eating them slowly because I mostly snack on them and I'm the only one in the family that likes pickled eggs so it can take me awhile to finish a large jar of pickled eggs.