I remember a similar rhyme from a nursery school in the mid 1960's:
I've got a secret I can't tell
Three little peanuts in a peanut shell
I can't tell, I can't tell,
three little peanuts in a peanut shell.

There were two additional versus, I don't remember the second, but the last was:
Why does the witch fly on her broom
back and forth in front of the moon
I know why, I know why,
to sweep the cobwebs out of the sky

I've been looking for the entire song for more than 30 years and am starting to think that this nursery school outside of Portland, OR may have written this tune themselves for the children there.