The best way to find out is to can some. Pickle them, put them in a jar with their vinegar (or fresh, if you want) and can then like you would do for anything else. Since they are eggs, I'd do them like a low-acid food. Pressure, not water bath. I dunno, try them both ways and see which one makes a pickled egg you can eat. They won't be BAD, but they might be too hard for eating. go for it.
Thanks for your opinion it always helps to talk to other canners. :) I was thinking that using my pressure canner and treating it as a low acid food would be the way to go so it’s good to know someone else had the same thoughts I did. I’m thinking that I can just mix the background put the hard boiled eggs into the jars and cover them with the beet mixture would allow them to pickle in the jar and save a few days for me.Now to find 2 quart jars that won’t cost a fortune to ship!