Answer: I have not read the book and so I am not sure if there is reference to a botanical name in it or not? That would of course be the most specific answer to your question.
I did however see the movie and in the movie, the fern is an unreal red!
Unfortunately, I am unable to find any references to a plant native to the Ozarks that has a common name of red fern and I have never seen one there, either.
There is a possibility that the fern in question could be the cinnamon fern since it is native in the Ozarks. This would be Osmunda cinnamomea. It is not red, but the new growth in the spring is a reddish brown and very striking. Perhaps that is what the author had in mind.
I'm sorry I can't be more specific.
Q&A Library Searching Tips