Sometimes they do. I always do.
Refrigerator: yes. My mother said you can take one day off the shelf life for every hour that eggs sit out. One of those rules of thumb that I simply never questioned.
You can easily write on an eggshell with a pencil. I write “HB” on the ones I don’t want to mistakenly break into a frying pan, and I mark the ones cracked while boiling so we can use them first.
Once I took an egg to work for lunch without checking for the “HB.” Got a messy surprise at my desk that day and felt like a total idiot.