Joshu's 'no' is a rejection of the question, not a response to it per se.

I know: Mu doesn't mean "no"; it means the "third answer", the something else left when you eliminate "yes" and "no".

Rather, I was answering "no" to the question, "Does a dog have a Buddha-nature?" to see what would happen.

What if the answer is "yes"? Why would saying "yes" make you lose your own Buddha-nature? I agree, the reason may be somewhere in how the Buddha-nature is different for the dog, it somehow negates our own Buddha-nature. So we choose our own Buddha-nature (we choose our context) and negate the dog's...but then that negates our own.

Analogously, either way you answer the question, "Do other people's moral beliefs have value?", you lose the value of your own. I suppose the power of the koan is that you can apply it to anything, like a torch pointed wherever you happen to already be looking. Or you can not focus it anywhere, and keep going with it, embracing "mu" indefinitely, becoming really, really wise...