What part of the cow would you consider they come from?
Giving you the benefit of the doubt, we looked up the word dairy in a dictionary to find that it comes from an Old English word for a "female bread baker." The root of the word is the modern-day word for "dough." And it is conceivable to us that eggs have as much relevance to the origin of the word dairy as butter does.
But nowadays, the word dairy only has to do with milk and cream and related products: sour cream, butter, cheese, yogurt, etc. You may often find eggs and dairy categorized together (several Ochef directory pages lump them together, for example), but they remain as distinct as the species from which they come.