"Eeny meeny miney mo" may not function as a counting song, but I'm told those first four words are indeed counting words, corresponding to "one, two, three, four" in an archaic British dialect (a system used for counting sheep, I believe). Perhaps historically-inclined UK correspondents will know more ...

Anyway, I assume that means some version of the rhyme predates the "original" racist second line, which I assume must have come from America?

Jonathan