Image from Anarchy Comics #1, 1978, edited by Jay Kinney.
For readers unfamiliar with the idiom: eat one’s words means retract what one has said, take back a statement, admit an error. So it’s similar to eating humble pie (whose origins are surprisingly visceral), and worth comparing with laughing on the other side of your face.
“You gotta break an omelet to make an egg”, of course, reverses the natural entropic order, playing with a proverb (“You can’t make an omelette without breaking eggs”) to make a political point. If you’re interested in the comic’s history, here’s a recent interview with Kinney at BoingBoing.