The definition at the "Whither Zither" website gives a good explanation of zipper verses.
"... zipper songs which are known as such because their lyrics for the most part don't vary from verse to verse except for a spot where replaceable words or phrases can be 'zipped' into and out of the song, like the animals in "Old MacDonald". Or like the activities in the song "Ain't it a Shame" which I learned years ago, along with many other wonderful zipper songs, from a Leadbelly's Last Sessions LP:
Ain't it a shame to [go fishin'] on a Sunday; ain't it a shame. (repeat) /Ain't it a shame to [go fishin'] on a Sunday, When you got Monday, Tuesday, Wednesday/ Oh, Thursday Friday Saturday, ain't it a shame.
In following verses, instead of "go fishin,'" Leadbelly sang "kiss your wife," "take a drink," and so forth. This is a late night road-song for me, and I take it down all sorts of provocative avenues of debauchery before I'm through.
....
In "Down by the Riverside," you can lay down all sorts of miscellaneous things by the river, along with swords and shields. "I Got Shoes" (you got shoes, all God's children got shoes)is also limitless. "He's Got the Whole World in His Hands" is a truly minimal zipper song. It's nothing but the title, repeated and repeated, with the phrase "whole world" as the zippable phrase, which returns in each verse for the last line:
He's got [the little bitty baby] in his hands (repeat three times) / He's got the whole world in his hands...."

"We Shall Overcome" and "She'll Be Coming round the Mountain" are others suggested.
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