Well, the song "Jerry" ("Mule on the Mountain") is somewhat older, a work chant. First collected I think by Zora Neale Hurston, who published it in her Mules and Men, 1935, p. 327, as "Mule On De Mount." First vrse:

Cap'n got a mule, mule on the Mount called Jerry (2)
I can ride, Lawd, Lawd, I can ride.
[OR He won't come down, Lawd, Lawd, he won't come down.]

Hurston calls it "The most widely distributed and best known of all Negro work songs." Subsequent collecting has not borne out that verdict, but the song is very widespread all the same, related to "Corn Bread and Molasses," "Coal-Black Woman" and other songs like "I Got a Bulldog." Some like the latter have penetrated into white tradition as well.

I have not heard the Josh White version, so I don't know how closely it resembles the above. But the name "Jerry" is usually associated with the above song. My guess, absent other evidence, is that Josh White may have derived his song from that source. At least it's a good place to start looking.

Bob