We lived on a boat in the Caribbean for two years, and we almost nevery refrigerated eggs. They were not refrigerated in any of the stores, either. IF eggs have not yet been refrigerated, they'll last a couple of weeks, or more, at room temperature. You need to turn the carton over every couple of days, so that the yolks don't settle against the sides, which gives them access to oxygen and speeds spoiling.
Please note that this is only for eggs that have NOT been previously refrigerated.
After a couple of weeks, if you have any doubts, drop the egg in a glass of water. If it floats to the top, it is spoiled (gasses inside cause it to rise). If it remains on the bottom, it's very fresh. If it hovers in the middle, it's old but still edible. Older eggs make the best hard-boiled eggs as they're easier to peel.
I hope this is helpful.
Stacey