I think I've heard it in old movies, by the type of people that would own a fainting couch.
Shouldn't it be: I'm not fine? I feel poorly is related to health alone.
That would be me non bene habeo. Note your translation has the negator "not," which can be translated into Latin. The use of the word "male" specifies that it should be translated as "poor" or "poorly".
The meaning may be similar, but on "I'm not fine" you are negating you're feeling fine, whole on "Me male habeo" you're making an affirmation that you feel poorly.
The word "poorly" is almost never used in the U.S., and when it is, it's only used as an adverb. (Oxford lists it as an adjective only in British English.) The opposite of "I feel well" is "I feel badly" when speaking of health. When speaking of general states, I would say, "I feel good" (or I'm doing well) and "I feel bad" (or I'm doing badly). I suggest adding "badly" and possibly "bad" to the acceptable responses.
Why isn't 'I feel ill' accepted? I understand 'poorly' but it's a bit archaic even in the UK, and rarely used where I live.
In Britain we often say I feel poorly. It sometimes means a bit under the weather, but can be an understatement!
I feel poorly is incorrect English unless you mean to say that one's ability to feel is below average.