A pun on boss, used here in the childish formula You’re not the boss of me! and also in Bruce Springsteen’s nickname The Boss.
The formula has been researched by linguabloggers, in particular by Neal Whitman, in two guest postings on The Volokh Conspiracy in 2004 (here and here), on recent uses; and more comprehensively by Ben Zimmer on Language Log, in 2007 and 2009. From the 2007 posting:
Several helpful readers pointed out the popularity of the phrase “the boss of me,” as in the song by They Might Be Giants, “Boss of Me,” used as the theme song for the FOX TV show “Malcolm in the Middle” from 2000 to 2006. “You’re not the boss of me now, and you’re not so big,” goes the chorus, seemingly from the perspective of a petulant child addressing a parent or older sibling. (Songwriter John Flansburgh has said it’s about his older brother.) Though the phrase “you’re not the boss of me” may owe some of its current popularity to the TMBG song, this bit of rebellious kid-speak has been kicking around since the late 19th century.
Specifically, back to 1883. In his later posting, Ben wrote about using Google Book Search for antedating, and finding an 1899 cite to add to the 1883 one.
In case you don’t know about Bruce Springsteen, here’s the Wikipedia intro:
Bruce Frederick Joseph Springsteen (born September 23, 1949), nicknamed “The Boss”, is an American singer-songwriter and multi-instrumentalist who records and tours with the E Street Band. Springsteen is widely known for his brand of heartland rock, poetic lyrics, Americana sentiments centered on his native New Jersey and his lengthy and energetic stage performances, with concerts from the 1970s to the present decade occasionally running at over 220 minutes.
Springsteen and his second wife, Patti Scialfa, have three children: Evan (born in 1990), Jessica (1991), and Samuel (1994). So the rebellious child in the cartoon could be Jessica as a girl.