Folk logic is when you conclude Sun revolves around the Earth, because that is how it seems to you. It is only logical since it is obviously moving across the sky and Earth is not going anywhere. You just don't go any deeper than that to find there is actually an entire science that deals with it.
In the same manner, the folk logic you used about animal rights is based with complete lack of awareness there exists an entire branch of philosophy called ethics, which has proven on many occasions that what ordinary people think is moral is actually not, since the conclusions of the ordinary people (folk) never go in depth and are satisfied with superficial explanations that only apparently suit them. We are all guilty of this, but we should try bettering ourselves.
Therefore, when you take an egg from the chicken, the chicken most likely won't miss the egg, but you nonetheless took something what isn't yours and this is theft, regardless of what you think is right, reality is you are a thief. Humans who are moral agents can decide whether to give something away, animals cannot, and therefore anything taken from them is theft.
As for cows, the only reason cows go into pain when they are not milked is because they are not supposed to be milked in the first place. I assume you do know why cows give milk? Why do for example humans and dogs or cats only have milk when they have babies while cows give milk all year round. The reasons behind it are quite immoral actually.