Eggs used to be classified as dairy, but since they don't fit the description of dairy, which means made from milk of an animal, they have been moved. They re made the food pyramid a few years back, and the 'dairy' group has become the 'milk' group, which does not include eggs. Eggs are now in the 'meat and beans' group, which is basically anything that includes protein. I think tofu is in there as well.
It is perfectly fine for vegetarians to eat meat, and in fact it's good because without meat you'll need to get the protein somehow.
However, if you are a vegan, eating eggs would go against that.