According to Aajonus Vonderplanitz refrigerated eggs loose nutrients. I don´t know if that´s true or not, he is a very controversial figure on this forum. Either way, after reading this I stopped refrigerating eggs. When I used to eat them raw I found refrigerated eggs easier to eat (tastewise) in the beginning of my initial raw food journey, but then I switched and left them out, even now in the summer heat (still inside though). But I think that when you eat them unrefrigerated they taste more intense, better and more "egg like". I´d recommend them to squirl inside a glass before eating, as gulping them down with the round egg yolk can be irritating (you tend to gag). Generally speaking I think that they could loose nutrients when cooled too much, I don´t think that our ancestors were refrigerating eggs, that came with modern times, neither do the hens.
Therefore I don´t think its too natural to cool certain foods. I think its best to consume them in their natural state as soon as possible (unless making high eggs). Since the egg comes with a shell I see no problem in leaving them outside for at least a month, even in summer. Even if eggs were expired I used to eat them raw. I think expiry dates are pretty useless anyway, unless its some commercial pre-packed food. If you leave them out for months they can become "high" eggs, full of bacteria, same as high meat. In some countries like china there are eggs which are left to rott for years, and they are a delicacy there. So I wouldn´t worry too much, however, I would worry if these are conventional eggs from cage hens. I would never touch that. Especially not raw.
Go for organic pastured eggs. Apparently fertile eggs are the best, which makes sense since they give the baby chick all the nutrients they needs for a healthy development, but these are hard to get. I would also go for eggs which are very orange as this can be an indication for the pasture they eat. Usually shitty caged eggs are very pale and yellow, unless of course they feed the chickens chemical colors which they do for farmed salmon. I ate many organic brands of different eggs and I can clearly say that the ones that had the most orange color were the tastiest. But the batches differed. One batch of the commercial organic brand was super orange and super tasty and then from the same brand some days later another batch had less quality, less taste and less orange color. I got super pissed and went to all the shops around to get from this previous batch. Then there are organic farmers which I know personally that have organic eggs, but they are less tasty and more yellow. And I asked them if these are pastures. Some where and some had organic corn to eat. I came to the conclusion that the pasture makes all the differene, think thats because of the beta carotene that makes the yolk orangish. Therefore I think when it comes to pastured eggs it all depends on the quality of the soil.