When going camping and bringing eggs there's not much you really have to do other than pack the eggs.
You can bring the eggs camping in either a cooler or just pack them in a box with the eggs left in the egg cartons.
I go camping for around 2 to 3 days at a time and sometimes for a whole week and when I do I just place the cartons of eggs in a box if I'm going tent camping and then place them in the bed of my truck.
The eggs do not really need refrigeration as long as you eat them within a few days to a week so they will be okay in the back of your truck or even in the trunk of your car.
When I go camping in my travel trailer I store the eggs in my camper refrigerator and then eat them before I leave camp.
It has never been a problem.
When camping I love to build a fire in a fire pit and use my cast iron cookware to cook eggs, sausage, bacon etc.