Yes, eggs can last a long time even without refrigeration. Ideally, get eggs that have not already been refrigerated (e.g. from a friend with hens), but I never do that since I don't have easy access to them.
An egg comes packaged by nature within quite a good bacterial barrier (the inner membrane of the shell), so do not disturb that membrane and ensure the egg shells are not cracked to start with. Gently cleaning the eggs would also be a good idea, but I use store-bought eggs which are already cleaned so do not bother. I pack eggs into one of those cheap plastic cases, lining both sides with paper towel. Smaller eggs tend to have thicker shells so avoid buying the extra large eggs (an egg farmer told me the eggs have a similar amount of shell around each egg, but sometimes it's stretched farther around a larger egg). If an egg cracks on a trip, I use it that day.
I took 2 dozen eggs on a canoe trip 2 weeks ago. Despite some very rough handling of the food barrel over multiple portages, not a single egg was broken or cracked before it's time.
There are some myths out there about eggs and how to preserve them. Here are some things you should NOT
- Do not crack the eggs from their shell and dump them into a bottle to take with you. This concoction that's created will certainly become contaminated with bacteria and will spoil quite quickly. If you want to go that route, buy the cartons of pasteurized eggs.
- Do not dip the eggs into boiling water in an effort to sterilize them. This only destroys that living membrane that is the real bacterial shield you want to preserve.
- Do not smother eggs in vaseline in a misguided effort to preserve them (not sure how that's supposed to work).
When space and weight are an issue, I use powdered eggs. The OvaEasy eggs previously mentioned get very good reviews and I've got them on my list of things to add to my kitchen kit.