The prohibition mentioned in the Torah is to mix meat (beef) with milk. This
injunction was extended by the Sages to include fowl as its meat can be confused with
beef. Eggs do not fall into this category as they cannot be mistaken for meat. Furthermore, eggs are considered a separate entity once they have been hatched and are considered to be pareve, or neutral, so that they can be
eaten with either milk or meat.
Interestingly, the Code of Jewish Law (Yoreh Deah 87:5) mentions that if fully formed eggs are found inside a chicken these may be consumed with milk products, however, if the eggs are not fully formed, they may not be consumed together with milk although one would be able to eat dairy products after eating one of them.