It's not SF, but it is a kids book and it's old enough (1951): Centerburg Tales by Robert McCloskey.
The chapter you're describing sounds like "Pie and Punch and You-Know-Whats" which has the following description on Wikipedia:
A new record on the jukebox in Uncle Ulysses's lunchroom has a song whose catchiness is such that nobody who hears it is able to stop singing.
Some bits of reviews and snippets on Google Books suggest the song is about donuts (and their holes) and that there is a simple trick used to undo the song.
TVTropes, of all places, has a longer description:
"Pie and Punch and You-Know-Whats" has someone put a horrible song on the jukebox in the lunch counter. Anyone who hears the song — whether the original jukebox tune or someone else's rendition — can't get it out of their head. Ultimately the main character gets it out of his head by using Punch, Brothers, then gives it to the rest of the town. Now he's cleared but they have it. So, he tells them to sing it to the one person who hadn't been in town. Now everyone is cleared except that person, who now has to be smuggled out of town to keep from reinfecting the whole town. The flip side of the same record causes the listener to get hiccups at the thought of the words "pie" or "Mississippi".