The question of Chicken eggs and why they are Pareve and not Meat has already been addressed. My question is why are they not considered "part of a live animal" and therefore prohibited to eat to Jews and Non- Jews alike?
Eggs are not considered eiver min hachai because the egg is not severed from the mother, but rather the egg is released from the mother naturally.
However if one struck a bird and this caused the egg to come out then the egg could be considered eiver min hachai if it is underdeveloped to the point of still being dependent on the mother to survive. In this case it is considered part of the mother's flesh and since it was caused to be removed by the action of striking, it is eiver.
On the other hand, if the egg was developed enough to survive (to hatching) on its own, it is considered independent from the mother and not part of her flesh and therefore, not eiver min hachai.
All of the above pertains to non-Jews and, not withstanding the Hatam Sofer, is the majority opinion.
The issue with Jewish consumption of eggs is more tricky. Not because of eiver min hachai but because of Kosher slaughter. To be considered kosher for consumption, birds must be slaughtered in the kosher manner. But eggs are collected from birds that have not had kosher slaughter. So the question is: Since the birds are not technically kosher to be eaten (they haven't had kosher slaughter) are the eggs which come from them kosher to be eaten?
For a more thorough treatment of the above check out either "Sefer Sheva Mitzvos Hashem" or the "Divine Code"; Both are by Rabbi Moshe Weiner.