The question, "does X
have the buddha nature
?" is, in a word, touchy
In my opinion, it can only be properly used in a mention context, not a use context. That is to say, you can talk about it, but you shouldn't ask it. Why?
Because the whole point of buddha nature is that it transcends labels, among other things, and to label something as having the buddha nature is to defeat it.
What should one do if a foolish acolyte asks this question?
This thought also applies to other questions, such as, does X have the funny nature?
The act of labelling something as funny takes away from its humor - although humor is sometimes twisted, making this very labelling funny...
But to take a complete joke, and then, as an outside label, not part of the joke, apply "funny" to it, weakens the joke. Therefore, when a foolish, young acolyte asks, "Is this funny?", the proper response is:
This short, metaphorical fugue courtesy of HandyDandy co.