You can trust the plastic case // sort of//
They are designed to prevent you from crushing the eggs while in a pack and are not really designed to keep eggs from breaking against the plastic while bouncing around on a moto bike.
Some stores will not have eggs in a 6-pack, just find a store that does and by two of them to take home. Those store shelf paper packs are great if you are smart enough to pack them correctly. Use parts of an old rag to add some 'stuffing' inside the case to keep them from bouncing, and then insert into a quart ziplock. Voila, you can carry eggs. Hardboil the other 6 for quickie snacks on the first two days of the trip.