At the beginning of the movie, after Vernon assigns them their essay topic, Brian is musing to himself,

BRIAN: Who do I think I am? Who are you? Who are you? … I am a walrus…

He's not just gibbering bizarre things to himself: He's quoting The Beatles. Specifically, he's quoting the song "I am the Walrus" written mainly by John Lennon (recall that Carl the Janitor says he wished he was John Lennon when he was a kid).

If you look at the lyrics to "I am the Walrus" they actually summarize the moral of The Breakfast Club—the idea that each one of the students transcends their stereotypes, sharing the experiences of "a brain, an athlete, a basket case, a princess, and a criminal" instead of just being limited to one of those identities.

The first lines of "I am the Walrus" run: "I am he as you are he as you are me / And we are all together." That's basically what everyone realizes at the end of the movie. And it makes sense that this is where Brian's train of thought is headed, because he's the one who actually writes the essay expressing this big realization on behalf of everybody else.

Also, this is just one example of an "I am the Walrus" reference in a John Hughes movie. In Ferris Bueller's Day Off, which he made the year after The Breakfast Club came out, the main character says,

"I quote John Lennon, 'I don't believe in Beatles, I just believe in me.' Good point there. After all, he was the walrus."

John Hughes: The guru of '80s teen comedies and the biggest Beatles nerd working in Hollywood.