Dog was considered an undesirable animal in the 17th century. So much so that there are a lot of phrases which refer to them negatively [tired as a dog, dog in the manger, down to the dogs, dog’s breakfast, dirty dog, etc.]. Sick as dog refers to being so sick that one may feel like vomiting. The first literary use of the expression is in 1705. The phrase still reflects in a negative sense as it was intended back then.