Place eggs in a saucepan and cover with cold water. Bring the water to a rolling boil and immediately cover and remove from heat. Let eggs stand in hot water for 17 minutes.
Transfer the eggs to an ice bath for 5 minutes. Peel the eggs under running cold water.
Place the eggs and beets in a large 1 quart jar, reserving the liquid from the beets for the brine.
Meanwhile, in a medium saucepan, combine the juices from the beets, vinegar, water, sweetener, and onion. Bring to a boil over medium high heat and boil for 1 minute.
Pour over eggs and beets.
Cover and chill overnight.
Notes
Don't overcook the eggs. Refer to this recipe for perfect hardboiled eggs.
The jar should be cool or at room temperature before sealing or pressure will build up once sealed.
Pickled beets and eggs should be kept refrigerated with the eggs and beets under the brine to prevent botulism.
Go easy on the beets if sticking to a strict low carb diet, be sure to watch your net carb count.