Does a dog have Buddha nature?

This is a Koan, a Zen question that is posed with no intellectual answer to train the student in intuitive awareness. Like many koans, the answer is intentionally illogical and non-rational. It is a common Ch'an/Zen method of attempting to stop short the questioner's thought process and "jar" their mind loose to have sudden enlightenment. He was in no way answering the question. The second you try to use words, language concepts or intellectualize to reach enlightenment, you are already way off. He used the question as an opportunity to implement this jarring method. Other such methods included shouts or being hit with a stick. As the koans became more popular, the masters began to despise them because monks would spend years studying them as a scholarly intellectual pursuit, priding them selves on having gleaned illusory knowledge from what was intentionally nonsensical. It is so far from the spirit and purpose of Ch'an/Zen. Ch'an/Zen strove to help people on the path to enlightenment by getting them to abandon concepts, language etc in favor of intuition. So any one who thinks they have an answer to one, is deluded.
As the story goes - while the master was meditating with his students, one of the students asked a question: "Does a dog have Buddha nature?" The most famous reported answer to the koan is "Mu!" Essentially this Japanese word could mean "Mindless.", "No mind.", maybe even "It doesn't matter". The "Mu!" could be directed at the questioner, the question or the answer. The answer is as much a koan as the question.
It should be noted that sentience (being self aware) is a prime requirement for a Buddha nature. People are sentient, amoebas are probably not. The location of a dog between these two points on this continuum is unknowable. Christians run into this exact question and problem when children ask if a pet dog went to heaven. Any logic based answer leads to unanswerable questions.