What to say to children that are not yours when they start to argue with you, and ask "why"?

I think you handled it exactly the right way. Most other kids would simply walk away if you told them "Because I'm his mom." This particular kid obviously comes from an enviroment with no discipline whatsoever, and he would have kept hounding you no matter what you said. Your consistency with your response eventually shut him off. His mom has a big job in front of her when he's a teenager, LOL!