OK, I know this recipe may sound strange, but this is a GREAT snack and wonderful sliced up on a salad. Easy to make. Easy to devour.
Put the vinegar, salt, water, salt, pickling spice and cinnamon (stick) into a saucepan and bring to a boil until the sugar dissolves. Take off the heat and pour in the juice from the beet can. Layer the eggs, beets and onions in a lidded container. When the liquid has cooled (warm is OK, just not boiling), pour over the eggs and beets, cover and put in to the refrigerator. Move everything around inside about twice a day, or you'll get white spots where the egg presses up against the container...not a disaster, but avoidable.
You can eat them at any point, but they are best after at least two days. I don't know how long they will keep because they've never lasted more than a week in our house.