Deviled eggs can be slippery little suckers. If you use a plate specifically designed for deviled eggs, they shouldn't slide around too much. If you use a normal plate, you can prevent them from sliding all over by adding some "roughage" to the plate. You can put spinach, shredded iceberg lettuce, or a bag of coleslaw mix on the plate. Then, put your eggs on top of the roughage to create a little bit of a bed to grip the bottoms of the eggs.