This is true, Pigs don't have sweat glands, which is one reason they roll in mud, to keep cool, another way they have of keeping cool, is to increase their rate of breathing (Panting), as for the saying, afraid I don't know.
Or maybe it from the old past-time of when a pig was covered in grease and chased through the village by young men who would attemt to capture it with rugby-style tacles..