This is the version we included in the Ballads and Old Songs chapter of the Rise Again Songbook. I found the song in a Revels songbook and in Jean Ritchie's Folk Songs of the Southern Appalachians.

There Was A Pig Went Out To Dig

There was a pig went out to dig
Chrisimas Day, Chrisimas Day
There was a pig went out to dig
On Chrisimas Day in the morning
(up) Am - - - / Am E Am E / 1st / Am E Am -

There was a cow went out to plough...
There was a sparrow went out to harrow...
There was drake went out to rake...
There was a crow went out to sow...
There was a sheep went out to reap...
There was a minnow went out to winnow…

trad. (English)

In Revels SB & Jean Ritchie FS of the So. Appalachs. On Revels Child's Xmas Revels, Jean Ritchie Carols for All Seasons, Sharon Lois & Bram Fam Xmas. Choral arr by Percy Grainger.

Here's a Revels recording: https://www.youtube.com/watch?v=-3YJJbXSpZc

And Jean Ritchie: https://www.youtube.com/watch?v=d-Wz4uXLHvY

And Percy Grainger's arrangement, by the Ambrosian Singers: https://www.youtube.com/watch?v=Ehqkf8CbN6Q

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The version used in Rise Again is an exact transcription of the lyrics from English County Songs (page 28), Lucy E. Broadwood and J. R. Maitland. Broadwood/Maitland got the words and tune from Miss M.H. Mason's Nursery Rhymes and Country Songs.
Notes from Broadwood/Maitland: "There are no words, properly speaking, after the first verse; but rhymes are invented according to the pleasure of the singer."

Don't know if this link will work for everyone:
https://play.google.com/books/reader?printsec=frontcover&output=reader&id=x_c4AQAAIAAJ&pg=GBS.PA28