It's all a scare tactic. Cancer (based on my bio 101 class once upon a time) is what happens when the wrong combination of switches go off in cells. They mutate into destructive cells that form tumors, destroy the body, ect. The list of things that people think cause that switch to go off is endless- UV rays, smoking, farting, you name it, someone's probably says it causes cancer.
That little clip is just silly. Anyone who has some knowledge in anatomy would know nails (and hair) are dead keratinized cells.This whole "nails need to breathe" thing is just ridonkulous. Nails for humans is hooves for animals. They nail metal things on horse hooves... does that cause horse hoof cancer?