negation - "Not I" or "Not me"?

After asking about “me” for answering a question. at Answering a question with "Me" instead of "I" + , I think I understand how to use “me” for answering a question.

However, I just remembered reading a children book, The Little Red Hen, many months ago and I found some sentences from that book.

“Who will help me to plant this wheat?” the little red hen asked her friends.

“Not I,” said the cat.
“Not I,” said the rat.
“Not I,” said the pig.

I don’t understand why those animals say “Not I.” I think they should say “Not me.”

Or is it that for speaking we should say "not me", but for writing "not I" is more appropriate?