I can't remember where I learned this, but it has worked for me for years. Instead of keeping eggs in the carton or even one of those plastic egg carriers made for camping, I crack my eggs into a leakproof Rubbermaid-type container for liquids with a screw top. I do this before I leave, and put the container in the ice chest or the pre-cooled TM fridge.
The eggs will stay separated and you can then pour out one whole egg at a time as needed, you never need to worry about broken or cracked eggs, and it saves valuable fridge or ice chest space.
The trick is not to break the yolk, so if you want to be extra sure that your eggs will pour out one at a time, you could break each egg into a measuring cup first and then pour it in the container. I can get about a dozen large eggs into a pint-size Rubbermaid container, which fits in the door shelves of my TM fridge.
I hope you find this helpful!
Betty