All true about the eggs.
Here in the US eggs are washed (mostly to get rid of barnyard/cloaca goo and the odd Salmonella) so the “natural water-and-germ-proofing” is gone, and we all refrigerate them. When backpacking sometimes I don’t worry about eggs being out of the fridge for the first day or two; they’ll be fine. If I want to have fresh eggs longer, or if I don’t care about the extra ounces, I’ll bring the cooler/blue ice described above.
But I always use the plastic egg protector with fresh eggs because it works much better in a pack than a hacked-off portion of a store carton. Have learned this from experience. The Coughlan carrier doesn’t crush (egg cartons do; I know), and it has little plastic fingers to hold the pointy end of the egg that provide a little bit of shock absorption.
Getting farm-fresh, unwashed eggs from some farmer is mostly a nonstarter for me, living in the middle of a city of 2.6+ million. I suppose I could look around for co-ops or the like, but I’ve never bothered. When hiking in other countries if desired I most definitely just get eggs from some local unrefrigerated source. But I do wash them with soapy water just before use and crack carefully.
Back OT: I used to scramble eggs when backpack cooking, but as mentioned several times up-thread, mixing the yolk with the whites does make things more sticky. These days I typically fry something =carefully= over my cheap, generic ISOButane stove in my thin, cheaply-adonized aluminum frypan, often using Ghee (Canadian Bacon is perfect for this; as can be that shelf-stable bacon, or maybe but likely not rehydrated freeze-dried hash browns), using lotsa swirling of the pan. Crack the eggs over the now done stuff, cover pan with lid/Alu foil, and low/slow-cook for 3-4 minutes adding a bit of water to steam the top of the egg yolk. Perfection! And little sticking/easy clean-up.