Why are milk and eggs considered non vegetarian?

Vegetarians who drink milk are called Lacto-vegetarians (that's how they want to classify themselves).

Milk is not vegan, however. Vegans eat no animal products at all.

Milk is produced for calves of Cows/Buffalos. So, thinking rationally you could only classify it as non-veg. Secondly they are proteins from living body, synthesized by living body and contains ANIMAL FAT.

Let us look at the composition of milk:

About 98% of milk fat is a mixture of triacyl glycerides. There are also neutral lipids, fat-soluble vitamins and pigments (e.g. carotene, which gives butter its yellow colour), sterols and waxes. Fats supply the body with a concentrated source of energy: oxidation of fat in the body yields 9 calories/g. Milk fat acts as a solvent for the fat-soluble vitamins A, D, E and K and also supplies essential fatty acids (linoleic, linolenic and arachidonic).

And most vegetarians say, animals are not harmed while harvesting milk so it could be called vegetarian. But its just a bad argument, if i suck 50 ml of blood from a goats body with a needle, not causing much pain or harm to the life of that goat and then drink it, will it make the blood vegetarian?

Its clearly non-vegetarian. And, anyone who call themselves vegetarian and drink milk could be called an hypocrite. We are just omnivorous social animals.