A very simple reason why eggs would be considered Pareve is based on the fact that chicken is only Besari m’derabanan, since people in the time of the Gemara were getting it mixed up with meat from animals. M’deoraisa, chicken is Pareve, since “lo sivashel g’di bachalev imo” can only apply to mammals – which produce milk – and not to birds. As such, when the Rabanan were Gozer on chicken, there was no g’zeira on the eggs, so they retained their Pareve status.
an Israeli Yid