I’ve been living outside the US for the past few years, and my contact with American culture has been primarily through TV and movies. One phrase I don’t remember hearing when I was there was “the boss of…” (e.g. “you’re not the boss of me”). The wording sounds really strange, and I don’t even remember “my boss” being a really catchy phrase either.
How did it get started?
As the oldest of three I heard this phrase a LOT when baby sitting my brother and sister. And this was, eh…12-18 years ago. It simply means “You’re not mom, YOU can’t tell me what to do!”
So they’d do it and I’d get in trouble when Mom DID come home. 
I’ve heard “you’re not the boss of me” for quite a while, but I’m sure where the exact phrase started from.
Certainly “bossy” has been said to kids for years, especially older siblings who are acting to dictitorial; e.g., “Don’t be so bossy, dear.” Then they heard that mommy or daddy has a person at work called a boss that tells them what to do. And oftentimes they complain about what their boss has asked them to do. So they associate bosses as being unpleasant people, because they’re bossy, which is something that they’ve been scolded for. So, when someone is being unfairly dictitorial, they’re pretending to be a boss, when they’re not.
I had a little brother, so I heard that all the time when I was younger.
If you can’t get the phrase out of your head, it’s probably due to the catchy theme song from the Fox comedy Malcolm in the Middle.
Pete
Long time RGMWer and ardent AOLer
Just seeing “the boss of…” made me laugh. With both older and younger sibs I heard and said that again and again. It has to come (as AWB said) from “bossy” and, “You can’t boss me around.” Thanks, curwin, for the reminder.