"Measles" in this context would be a disease of swine and cattle, caused by larval tapeworms; these encyst in the tissue and render the meat unusable.  Sometimes, presumably, the animal dies of the infection.

An unpleasant thought.  The song -or rather, just the one verse- is also well-known in the UK, and often sung with, I imagine, no thought that anything other than the other "measles" (the childhood disease, more common than Rubella or "German" measles) is meant.