I don't think its anything to be worried about, since you say this isn't a permanent thing. In fact, before the industrial revolution, when we didn't have all sorts of food, seasonal and not, at arm's reach all year round, people would rely on whatever was available, and made the most of it.
Eggs are definitely a great choice; an egg is a complete source of nutrition, it provides all the nutrients needed for the embryo to develop. Eggs are a great source of choline, selenium, iodine, B vitamins, protein - as you remarked, molybdenum, phosphorus, vitamin B5, vitamin B12 and vitamin D; as well as antioxidant carotenoids (lutein & zeaxanthin) pastured eggs will have a better nutritional profile, with a more favourable omega 3:6 ratio, and more vitamin D since they have more access to the outdoors; they have a greater concentration of antioxidants (and hence grass-fed eggs usually have brighter coloured yolks).
Eggs per se are not in any way a problem - the only issue is the avidin in the white, but as long as you cook it, you neutralise that. You could even stick to the yolks alone if you wanted, they have all the nutrition in the egg (the white is really just protein). Of course, if you continued this indefinitely, it could be an issue just as it would be in the case of restricting your protein intake to any one source, be it eggs or beef or anything else, especially if you don't have a constant supply of pastured eggs.
Besides, its not like eggs are the one and only thing you eat. Choose the best eggs you can get, preferably pastured, if not, organic and free-range; do the best with what you have; omelettes are also a great vehicle for nutritious fillings, such as vegetables, even fruit (flaugnarde, anyone?), as well as meat, fish, and cheeses, and you'll be getting adequate nutrition; especially since you don't plan to do this forever: all foods, really, are seasonal; eggs are probably at their best in spring and summer when they will naturally have better nutritional profiles, and variety is always the best way to get maximum nutrients.