CorusKate It's okay. The 'original' wording, isn't actually the 'original'. Eenie, Meenie isn't inherently racist. The racist version is believed to have developed in America, and happens to have been popularised in print in the early 20th century. But it is not the 'proper' version.
Just use the version you liked when you were a child, or spend your life trying to research which of the early 19th century versions came first. But don't let the racism of some New York children in the 1888 taint your memories!
Hey, here's a Cornish version from 1882:
Ena, mena, mona, mite,
Bascalora, bora, bite,
Hugga, bucca, bau,
Eggs, butter, cheese, bread.
Stick, stock, stone dead - OUT
en.wikipedia.org/wiki/Eeny,_meeny,_miny,_moe