A new man joins the cowboys, and proves expert on many things. The cowboys think he must be a greenhorn, and allow him to take on the wild Zebra Dun. To their surprise, he controls the horse and receives a job. Not all educated people are greenhorns...
Larkin states, without evidence though it's a reasonable conjecture, that the horse in this piece was a dun with the Z bar brand, with the "Z bar dun" wearing down to the "Zebra dun." Her other conjecture, that the singer may have been an Englishman who learned to ride while hunting fox, seems much less likely.
Logsdon (who mentions Larkin's suggestion) notes that this is because, while "dun" is a recognized description of a horse's hide, "zebra" isn't. But he observes that a "zebra dun" would be a dun with stripes.
One of Logsdon's texts, interestingly, was sung to the tune of "The Battle Hymn of the Republic"/"John Brown's Body," though there is no chorus and some of his lines must have taken some quick tongue-work to sing to the Battle Hymn. - RBW