Place egg cartons on refrigerator shelves lengthwise, front to back and they will never fall out. Even if they vibrate toward the door, the door will stop them from tumbling out due to their length.
Usually, the carton lid will protect the eggs if anything falls in the fridge while driving. Use the eggs from the rear of the carton first. This will provide stability when you lift the egg carton. Use some empty egg cartons as separators in your fridge. They will buffer and protect items.