Personally, I'd never thought about it that way, but I can definitely see where she's coming from!

Below is the text which Mike Yates references. Personally, I don't think it's quite right to say the text here is per se Cluck Old Hen; it seems to me more either an antecedent or a partial fragment. It does share a number of verses which are often combined to make the chorus, but doesn't have the titular "Cluck Old Hen" lines that you'd expect to see. I think this is a case of some floating verses being shared. It to me bares some resemblance even to the family of lyrics related to Rise When the Rooster Crows -- the ones about the ducks chewing tobacco and the rooster keeping time (see here or here for example).

I'll include a few relevant excerpts from the text. It's worth noting that the date is approximate and, from the notes of other songs in the chapter, seems to be recalled from Gates's memory rather than written.

Source - Rainbow In the Morning, ed. J. Frank Dobie: "South Texas Negro Work Songs" by Gates Thomas (1926, re. 1965) pp. 157, 159-160:

After each song, I have put a date indicating approximately when I found it. I have indicated whether I know or conjecture of each song's history or regional meaning, with such references to analogues or cognates as I have had time to work up. Both notes and cross-references are incomplete. Titles of the anthologies consulted are as follow:

J. H. Cox: Folk-Songs of the South, Harvard Press, 1925.
John A. Lomax: Cowboy Songs, Macmillan (Rev. Ed.), 1916.
R. E. Kennedy: Mellows, A. & C. Boni, 1925.
Odum, H. W., and Johnson, G. B.: The Negro and His Songs, University of North Carolina Press, 1925.
Dorothy Scarborough: On the Trail of Negro Folk-Songs, Harvard Press, 1925.
T. W. Talley: Negro Folk Rhymes, Macmillan, 1922.

...

(2) THE OLD HEN CACKLE

The old hen she cackle, she cackle in the corn;
The next time she cackle, she cackle in the barn.

Chorus:
Well, the old hen she cackle, she sholy gwain to lay.

The old hen she cackle, she cackle in the loft;
The next time she cackle, she cackle further off.

Chorus:
Well, the old hen she cackle, she sholy must-a laid.

The old hen she cackle, she cackle in the lot;
Well, the next time she cackle, she'll cackle in the pot.

Chorus:
The old hen she cackle, well, she sholy ought to lay.

--- 1886

No. 2 is one of the liveliest dance tunes on the list, working into the organic melody of the hen cackling. For analogues, see Talley, 50, 93.

Also worth noting is that a song of the same name ("Old Hen Cackled") appeared in a 1915 list of violin tunes, The Berea Tune Lists: An Archival Resource for the Study of Social Music in Eastern Kentucky and East Tennessee in 1915