The earliest version I found was credited to Emilie Poulsson's Holiday Songs, a book which was referenced in many indexes about kindergarten curriculum as simply "HS." Annie L. Preston is listed as the composer and this was published in 1912.
Found on page 3 of Games and Dances: A Selected Collection of Games, Song-games and Dances Suitable for Schools, Playgrounds, Gymnastic Associations, Boys' & Girls' Clubs, Etc By William Albin Stecher · 1916
In the preface the author thanks the Milton Bradley company for permission to use the song. It may have been recorded on a record, but I was unable to find it. UPDATE: I found the earlier source, Holiday Songs, that was published by Milton Bradley.
The other major source where I found this song was the Frank C. Brown Collection of North Carolina Folklore. Browsing through this collection you will find many other singing games that are found in a variety of other music education core texts. "Let Us Chase the Squirrel" was collected in 1927.
I found two sources that says there's a recording on Victor Records #17568. I found it on Ebay, but it was only the sleeve.