Well, if refrigerating pickled eggs was necessary, there would be a lot of sick people leaving bars on a Saturday night. I cannot tell you how many taverns I have seen with a big jar of pickled eggs behind the bar. Heck, we used to have bets that nobody could eat 5 in a minute in my younger wilder days.
I would think that you have to refrigerate them during the brining process, but after that, the high acidity and salt content keeps bacteria at bay. According to Wiki, no refrigeration is needed:
Pickled egg - Wikipedia, the free encyclopedia
Pickled eggs are also popular with campers because they do not require refrigeration once they have been prepared.
This does bring back memories for me.. My mother used to make pickled eggs when I was a kid from the brine left over from a pickle jar. We would get quite a talkin to if we threw out the pickle jar/brine when all the pickles were gone.
I will say this other note on refrigeration, pickled eggs in my opinion taste better cold, so if you are basing it on that, I would say, put them in the fridge.