We still don't know who wrote it. Some of these songs in college songbooks were written by alumni or students anonymously, authorship has been determind for some.
Couldn't find in American Memory or Levy. Doesn't seem to be related to any of the "Rigs" songs, of which there are many.
Not found in W. W. Newell, 1883, 1903 and later reprints, "Games and Songs of American Children." His song collecting was confined to a few cities.
I haven't checked other game song books except to glance at Gomme and Opie. Not found, but could be under a variant title, or could be an American development. There are a couple of UK games, including "Spanish Ladies," which have the first line, "As I was walking down the street."