We have taken fresh eggs for many years with no problems, except for occasional breakage.  We buy medium eggs in a styrofoam carton (the last thing we buy just before our departure), double wrap the carton in two brown paper grocery bags and duct tape it underneath the stern seat of the canoe.  It stays in place while portaging.

In camp we always are careful to set the egg bag in a shady spot, and we plan to use the eggs within 2-3 days if the temperatures are in the 80's or above during the day.  During cool weeks they last longer, planning two or three per day.  There are just two of us, so we only take one dozen and then use BackPackers Pantry Denver Omelette for our remaining breakfasts.

On the rare occasion when breakage occurs we have an omelette that evening!

My daughter in Chicago can get pasteurized eggs and I am sure they would be safer, but I haven't seen these in the grocery stores at Grand Marais or Ely.