I stumbled upon this (the song, not the cat) while perusing a Civl War Song site.
America Singing: Nineteenth-Century Song Sheets

WHITE CAT AND BLACK CAT (George S. Harris' card and job printing office. S. E. corner of 4th and vine Sts, Philad. [n. d.])

White cat, black cat, any cat at all,
When you catch de pussy cat don't you let her squall.

CHORUS.

When you catch de white cat, sabe him, oh, sabe him,
When you catch de black cat shabe him by de tail.
Tom cat's a gemmen an' he rambles in de park,
Bull dog de bow-wow, skeer him wid his bark.

Ole cat and kittens was playing on de bay,
Big cat get out ob de little cat's way.

When you catch, etc.

Tom cat, he bellows at de bristles on his back,
'Kase he see de pussy cat a peepin' thro' de crack.

When you catch, etc.

Green peach puddin, and a punkin pie,
De black cat kicked out de white cat's eye.

When you catch, etc.

Big cat, little cat, any sort o' cat,
Skit at te at, te skat, te it, at de skat.
When you catch, etc.

PRINTED AND SOLD WHOLESALE AT GEORGE S HARRIS' CARD & JOB PRINTING OFFICE, S. E. Cor. 4th & Vine Sts. Philad.

Stamped: 1490

COLLECTION: American Song Sheets

REPOSITORY: Rare Book and Special Collections Division, Library of Congress

DIGITAL ID: as114900