That one sounds weird to but I think it may have a little more credability. I think it came from people with horses. Horses usualy do not lay down unless they are sick, hurt, or dead. So when you put a horse down it actualy falls down. There is also an expression that a horse is down. In this case "down" meens laying down sick.
On the lighter side, there are many euphamisms for defecating and urinating. I think the worst is, "I have to powder my nose." I think someone is powdering the wrong end.