Well, it sounds highly unlikely to me. Where did you hear such a peculiar thing? Presumably no king would *want* to be called John, because of the unhappy associations (King John is traditionally regarded as a Weak Man but a Bad King, or possibly vice versa), but that's not the same. Prince Charles, similarly, apparently does not wish to be known as Charles III when he becomes king, because of the associations with Charles I. Of course, such concerns are pretty pointless because the average British person wouldn't have the faintest idea who Charles I was or what he did, far less King John.
Now, if a king were *Catholic*, that would be a whole different matter...