Let us finally settle this debate.
To understand and critically analyze this topic one must understand what the practice of Vegetarianism: it is to abstain from the consumption of meat.
The issues with this topic relies on the grey line drawn around the consumption of animal by-products. While many individuals are comfortable consuming milk, cheese and honey, the consumption of rennet (sourced from stomach lining of calves) and gelatin (sourced from bones, skin, and connective tissues of pigs and cows) is largely associated with non-vegetarians.
Eggs however are a different ball-game.
We have come a long way since the proto-chickens had an existence. These animals were domesticated and subjugated to selective breeding, where only certain hens were allowed to mate and pass on their genes to the next generation.
This ensured that the chickens we have today are hens who have a natural capacity to lay more eggs, fertilized or not. Thus due to selective breeding, we had chosen egg-making factories as today's "normalized" chickens.
The argument that eggs are vegetarian states that because there isn't any meat, bones, skin or connective tissues in an unfertilized egg, it is to be considered vegetarian.
We have indeed come a long way since chickens have been laying eggs. While the wild chickens (if they do exist) may still lay fertilized eggs majority of the time, today, our farm-grown modern eggs are unfertilized—Vegetarian Eggs.
Some may have reservations when they see a tiny blood spot on the egg yolk. This isn't a connective tissue or signs of a fertilized egg. This sometimes occurs when a blood vessel bursts upon laying the egg. It is natural and safe to eat.
This settles the debate. Eggs weren't vegetarian. They used to be non-vegetarian a 1000 years ago. However, due to human interventions, specifically animal husbandry, we have converted eggs from being non-vegetarian to vegetarian. Some may even call eggs vegan— but, that's a debate that deserves its own page.