This quote appears in the closing lines of Part One of Fahrenheit 451 when Montag picks up a book at random and starts to read aloud to Mildred. The quote comes from the book Gulliver's Travels by Jonathan Swift, and tells the story of how the two great empires Lilliput and Blefescu came to be at war. The conflict arose over the question of which end to break one's eggs: those in Blefescu break their eggs at the big end, as tradition dictates, but those in Lilliput are forced to break their eggs at the smaller end as a result of royal edict. There are, however, rebels in Lilliput who refuse to break their eggs this way, which eventually causes a war, as Montag learns when he reads this section:

"It is computed that eleven thousand persons have at several times suffered death rather than submit to break their eggs at the smaller end."

This quote is significant because it encourages Montag's growing sense of rebellion. It is like a call to action because it suggests that if people are prepared to sacrifice their lives over something as trivial as the breaking of an egg then Montag ought to stand up against the serious problem of censorship. While Montag does not understand the quote at the time of reading (he asks Mildred what it means), it clearly has an impact on his subconscious because he continues reading the books and later seeks the help of the retired professor, Faber.

Please take a look at the second reference link for more information on the war between Lilliput and Blefescu.