Place the eggs in a medium saucepan and cover them with cold water. Bring to a boil over medium heat.
Once boiling, reduce the heat to low, cover the pan, and let the eggs simmer for 10 minutes.
After 10 minutes, remove the eggs from the heat and place them in an ice bath to cool for about 5 minutes. Once cooled, carefully peel the eggs.
In a mixing bowl, whisk together the beet juice, water, white vinegar, sugar, and salt until the sugar is dissolved.
If using, add the mustard seeds, black peppercorns, garlic powder, and bay leaf to the mixture.
In a clean jar, place the peeled eggs and pour the beet juice mixture over them, ensuring the eggs are fully submerged.
Seal the jar with a lid and refrigerate for at least 24 hours for the flavors to develop.
Serve chilled as a colorful appetizer or a tangy side dish.