inefficient or no, nothing beats a fresh fried egg in the morning, or french toast made with fresh eggs.
I normally use one of the 12egg carriers, some of these are lighter than other, depends on the plastic. Although for larger quantities, wrapped in newspaper works stuff inside anything you have. I bet you could make a white nylon stuff-sac and wrap them in newspaper, and securely fasten the sack to the outside of your pack for warmer times..
Here's one big big trick though. If you want them to last, buy non-refrigerated farm fresh eggs. If you're old enough, you'll recall the days that people didn't really refrigerate them in the first place, just like butter on the counter.
Once they've been refrigerated they seem to go bad faster once removed from that environment. That's also the only way I've ever known anybody to get salmonella from eating raw eggs.
Mayonnaise is the same way, if you don't refrigerate it in the first place, it isn't nearly as likely to spoil at warmer temps.