Or, more specifically:
If you finish that question, you will have answered it.
Do you know why I say this?
First, I shall clarify the question for those who do not know what Buddha nature is. The question asks, is a dog valuable? Is it a good thing? What is it worth?
The question cannot be answered because it violates the very nature of worth. It assumes that Beauty is a property intrinsic to an object. While the nature of the object surely affects the Beauty perceived in it, this is not the complete picture. You may have heard the old saying, "Beauty is in the eye of the beholder." This is partly true. Certainly, the identity of the beholder can affect the level of Beauty to be found. But even knowing the subject and the object is not enough to form an evaluation of Beauty. They miss a vitally important part of the interaction: The context within which it takes place; the actual event that brings about the perception of Beauty. (For Beauty is more akin to an event than to a property. In a sense, it describes the nature of any interaction between our selves and the world, as it enters our minds. But you do not need to understand this in order to see the trifold requirements for a judgment of Beauty.) Subject, object, context. (Really, context contains subject and object, but they are consistently important enough to deserve separate mention.) Let us try filling these in in various ways: I want my slippers, and the dog fetches them for me without damaging them or slobbering too much on them. This event, to me, has Buddha nature. Ralph, who has a severe phobia of dogs, wants his slippers, and the dog brings them to him; he spends the next five minutes frantically trying to get away from it. To Ralph, this event lacks Buddha nature. Ralph is my worst enemy, and thanks to the five minutes of distraction, I gain the upper hand in our adversarial conflict. To me, his dog/slipper event has Buddha nature. Now let's say a tiger brings me my slippers. As much as I admire tigers from afar, I admit that seeing one randomly walk into my living room would freak me out, slippers or not. Lack of Buddha nature for me, plenty of Buddha nature for the joker with the hidden camera and the nationwide audience next week.
Now go back and see why "Does a dog have Buddha nature?" is not answerable.