It's not from "Buckeye Jim," pretty certainly, since it doesn't appear in the only version of that song ever found -- it was collected by Fletcher Collins in North Carolina and subsequently popularized by Burl Ives. (It's a version of the old "Limber Jim" riverboat song.)

The form of the verse, and the phrase "not far off," suggests "Gonna Have 'Lasses In the Morning," as done by the Golden Melody Boys back in the 1930s.

Way down yonder, and it's not far off,
Where the jaybird died of the whooping cough,
They eat snail puddin', pennywinkle pie,
The grey cat kick out the black cat's eye,

CHO Oh me, oh my,
         Gonna have lasses in the mornin'.

But since the second half of the verse isn't the same, I have to guess it might be a version of one of the "Kemo Kimo" songs, or else perhaps a version of "Down in the Arkansaw."

Problem is, that jaybird couplet is a fairly common floating verse and turns up in many songs like "Raccoon and Possum," "Pretty Little Turtle Dove," etc. There's also "Saturday Night and Sunday Too," with the verse:

Jaybird pulled a four horse plow, sparrow why can't you?
'Cause my legs is little and long, scared they'd break in two.

There are plenty of good traditional jaybird verses around if you want to put together a Jaybird Song.   

Bob