I doubt if there's any country where none of the population eat eggs. But there are many groups of people throughout the world who don't.
Early Christians were forbidden to eat eggs during Lent. Strict vegetarians regard eggs as meat and so won't eat them (e.g. many Hindus). In Rajisthan, a holy city in India it is illegal to eat eggs and meat within the city. In other countries, like parts of Africa, various people traditionally aren't allowed them for cultural reasons. The list includes pregnant women and circumcised girls.
Apparently, there's still some byelaw around that makes it illegal to eat eggs in Ney York at the weekend!