"Punk" was a youth movement started in the 1970s (I believe). It's characterized (generally-speaking) by rebellion, anarchy, mild body-modification (ear- and nose-rings), and just generally outrageous and conspicuous appearance in clothing and hairstyle. The music is also rocky, anarchic, profane, and loud (and doesn't always follow the conventions of harmony and melody laid down during the era of common practice; although some of the Sex Pistols' music is pretty musical, in my opinion). These days "punk" and "anarchy" are more or less synonymous, hence the message on the pillow.
"I'm too rebellious, anarchic, free-spirited, etc. to agree with anything you say." Taken to that extreme, it's irrational. :)