It depends on people lifestyle, religious views and where you live.
If you are a vegetarian because of cultural and religious views, egg is non veg.
If you go to any North American or European restaurant, a vegetarian dish does not mean it does not have eggs. Even on flights when you say vegetarian it does not mean that your food will be egg free.
I am a Lacto-vegetarian (no meat, no eggs, no lard, no seafood, no rennet, but ok with milk). In certain restaurants I usually say I am vegan (no meat, no eggs, no lard, no seafood, no rennet, no milk) to be sure that my food is strictly vegetarian. On flights, asian vegetarian would mean - no meat, no eggs, no lard, no seafood, no rennet, but ok with milk.
Then there is the term vegetarian egg, which can mislead people. The hen who laid the egg was fed a vegetarian diet. I always find this one funny! The egg is still non-veg though!