In olden times, pubs put up signs so they could be distinguished by a largely illiterate public. So they'd have a sign showing a swan, or a red lion, etc. A boar would also have been a common sign, too. Presumably someone had the bright idea of painting the boar blue to make it stand out, (or they didn't have any other paint!),and so the Blue Boar came about. Supposition, I know, but it seems the simplest explanation.
With respect to heathfield, it actually all began with a man in the Midlands who told interminable stories, invariably dirty, and using the most dreadful language. People had a Hell of a job getting away from him, and he became known as the Blue Bore. It caught on because there's one in most pubs, and they couldn't spell back then