In a small saucepan add the beet juice, vinegar, water, sugar, salt, and pickling spice. Bring the pickling brine to a simmer for ten minutes, or until the sugar and salt have completely dissolved.
Remove the saucepan from heat and let the brine cool to room temperature.
Add boiled eggs to a large mason jar and pour in cooled brine. Let the eggs pickle for 24 hours to 1 week. After 24 hours there will be a purple ring on the exterior of the eggs, but they will be white inside. After one week they will be fully dyed.
Once your eggs are pickled, slice them in half lengthwise with a sharp knife and remove the yolks.
Mash yolks in a small bowl with the mayo, sour cream, Dijon, lemon juice, horseradish, chives, salt, and black pepper.
Place the deviled egg filling back into the sliced eggs and serve.
Notes
If you have room in your mason jar, you can add chunks of the canned beets for additional flavor along with the eggs.