Pickled Eggs - Include Sugar & Salt?

We make lots of pickled beets using the Ball Blue Book recipe. The vinegar is diluted & we reduce the sugar quite a bit. No salt. I hard boil eggs, peel them &:put them in a container & add enough pickled beets to cover. After a day or two in the fridge they are classic Red-Beet Eggs. The salt & sugar are not essential to the pickling process & can be adjusted or eliminated to suit your taste.