Hard-boil the eggs: Place the eggs in a large pot and cover with cold water by about an inch. Bring the water to a rolling boil, then remove from heat, cover the pot, and let the eggs sit for 10-12 minutes. Immediately transfer the eggs to an ice bath to cool completely.
Peel the eggs: Gently tap each egg on a hard surface to crack the shell all over. Carefully peel away the shell under running cold water.
Prepare the pickling liquid: In a medium saucepan, combine the undrained beets (including the beet juice), apple cider vinegar, sugar, water, salt, peppercorns, cloves, and bay leaf. Bring to a boil over medium heat, stirring until the sugar is dissolved. Reduce heat and simmer for 5 minutes.
Pack the eggs and beets: In a large, clean jar or container with a tight-fitting lid, layer the peeled hard-boiled eggs and beets. You can leave the beets whole if they are small or cut them into smaller pieces.
Pour over the pickling liquid: Pour the hot pickling liquid over the eggs and beets, ensuring that the eggs are fully submerged.
Chill: Cover the jar or container and refrigerate for at least 24 hours, or preferably longer (up to a week for best flavor). The longer they sit, the more intensely pink and flavorful the eggs will become.