Easiest way in my opinion is to break them into a zip lock, beat them soundly to mix well, and freeze them solidly. You also have the added advantage of another Ice pack (sort of) for the cooler.
I have had them last 2 weeks in this form, never had a reason to push beyond that, cause, you know, we couldn't keep anything frozen longer than that, and had little reason to go longer anyway.
My logic was bringing eggs on a trip is tatamount to challenging the river Gods; like you were daring them to break your carefully packed eggs. But, If I did it myself, the River Gods were appeased.
It is correct that you have to beat the yolks into submission; once frozen they are reluctant to take any other shape or play nice with any other ingredient. Kind of Zombie like for some reason.
The only ones who ever complained were those who expected their eggs sunny side up, or over easy. But, if they couldn't deal with scrambled, then we generally had bigger issues with them than broken eggs. Like, "why the hell is there sand in my beer?", and "what do you mean, I have to blow up my own air mattress?"
Schutzie's advice; break em and freeze em; the river Gods have enough opportunity to fuck with you already.