Milk is not a juice and has many properties that would go against an authentic juice fast.
You will also get many opinions on milk. I find that most people that have studied for a while in the field of health will conclude that milk is not a healthy substance. It is meant to mature calves into cows. Humans require human milk for a short time to do the same. They are very different in their properties.
Can you imagine how crazy the first human being was that got down on their knees and suckled a cow for the first time? I think you get the point as to how natural the practice actually is. People have done it for so long they don;t connect the absurdity of the practice.