For the beet-pickled eggs: Put the cold eggs in a medium saucepan and cover with cold water. Heat the water to a boil, then cover the saucepan and remove it from the heat. Let stand for 12 minutes. Drain the eggs and rinse with cold water. Let the eggs cool a bit, then peel them while they are still warm (eggs are much easier to peel when warm).
Place the hard-boiled eggs in a bowl or storage container and pour the pickling brine from the beet jars over the eggs, making sure the eggs are submerged. (You may need to place a weight or a plate on top.) Allow the eggs to sit for 45 minutes to 1 hour in the brine, but no longer or they'll start to get rubbery.
For the sweet Hungarian filling: Mix together the mayonnaise, sour cream, dill, vinegar, horseradish, mustard, paprika and cucumber. Season with salt and black pepper.
Slice the eggs in half lengthwise. Gently remove the yolks by pressing your thumb against the back of each yolk to pop it out of the white. Add the yolks to the bowl with the filling and mash with a fork until smooth. Season with salt.
Spoon the filling into the egg whites, or use a re-sealable plastic bag to pipe it in. Sprinkle with paprika and dill before serving.