When buying store bought eggs that have been refrigerated before you bought them and brought them home then yes the eggs need refrigeration especially if you're not gonna use them within 1 or 2 days.
However there have been times I've left out eggs on the counter for 5 days before and they were still okay to eat and haven't gone bad.
But if possible always keep the store bought eggs refrigerated when not eating them or using them for cooking etc.
The refrigeration will make the eggs last longer than they would being outside at room temperature.
Farm Fresh eggs usually are okay out of the refrigerator for a few weeks or so but it's always recommended to keep eggs refrigerated.