The "green sickness" aka chlorosis explanation has to be urban myth. Anaemia doesn't make you green, it only makes you less pink - it was given the name "green sickness" by wild exaggeration in the 19th century. Nobody in the Middle Ages would have made that connection.

There are peoples in Amazonia whose traditional way of life discoloured their skin with green mould, the same sort of biology that makes sloths green. There might be a remote chance that was involved.