I may have found the song Louise was looking for: it looks like a variation of "William a-Trimbletoe" or William Trimbletoe, part of a counting game.

Here's a version collected in Texas:

William a-Trimbletoe,
He's a good fisherman;
He catches hens and puts them in pens,
Some lay eggs, and some lay none
White foot, speckled foot, trip and begone.
Wire, briar, limber lock,
Three geese in a flock,
Sit and sing till daylight spring,
Till comes the Tod with his long rod,
To chase the Czar, to wig, to war.
Ellison, dear, come lend me a spear,
To go in the woods and kill the king's dear.
Hetom, petom, potom, pie,
Popular, gigama, gi

From Texas and Southwestern Lore, p. 67