Terminology: when I refer to "canned eggs", I'm referring to pickled/marinated eggs that have been processed, and can be stored at room temperature long term (i.e. shelf stable). When I refer to just "pickled eggs", I'm referring to pickled/marinated eggs that have been processed, but require refrigeration.
I've been pickling eggs for years now (eggs plus hot peppers and garlic plus a tiny bit of beet for red coloring), and a neighbor recently gave me a large amount as she can't sell them at the local market for the next several weeks (i.e. hundreds of eggs), so she gave them to me. I'm planning on making several dozen jars of them. I've used the process described by the "Kuntz Family Picked Eggs" article I found online years back, and never had any problems.
However, the page, citing the NCHFP, notes that it's not recommended to attempt canned (i.e. shelf stable and no refrigeration) pickled eggs. It goes on to describe a recipe that should be safe for canned eggs, provided they are kept in a dark, cool location. I've also found a lengthy StackExchange:Cooking discussion that arrives at the same conclusion: canning eggs is a no-go.
I've recently discovered a brand of pickled eggs, "Nanjo's", that is sold at the local grocery stores, and it's' canned pickled eggs (room temperature, no refrigeration). They also taste quite good.
So, is there some trick to making shelf stable pickled eggs that I don't know about (i.e. pressure canning versus just hot water boiling) that makes them guaranteed shelf stable (i.e. by a reputable authority on the subject) and safe to store (i.e. at least for 6 months) without refrigeration? I'm skeptical that such a product could be sold in major stores if it had a serious risk of botulism contamination.
I found some posts by the author of the first web site I linked: