noun /ˌvejiˈte(ə)rēən/A person who does not eat meat, and sometimes other animal products, esp. for moral, religious, or health reasons
adjective /ˌvejiˈte(ə)rēən/Of or relating to the exclusion of meat or other animal products from the diet
- a vegetarian restaurant
There are plenty more definitions for the word on the internet, but the general meaning seems to be "exclusion of meat and fish from the diet".
So why are eggs included the non-vegetarian section? It is neither meat nor fish. It is an animal product such as milk and honey which are completely acceptable to vegetarians. Eggs being non-vegetarian is blindly accepted without thinking why.
The common reasoning seems to be that eggs are non-vegetarian because it has potential for life.
Chicken eggs are unfertilized when they're laid. Hens have the ability to lay eggs without any contact with the male bird. Therefore an unfertilized egg contains an embryo which under the right circumstances could generate life. Eggs meant for consumption do not meet these circumstances. No life is killed.
A seed could become a tree, but not every seed gets planted.
When you look at the contents of the egg, the white is 10% proteins in water and the yolk contains fat and cholesterol needed to nourish the embryo. Again, it is like a seed that holds the nutrients needed for a plant to develop.
There may be moral or ethical reasons to not eat eggs (such as inhumane chicken-rearing methods) or you might just not like the taste. But by definition, eating eggs does not make you a non-vegetarian.