Er, no, I think not. It’s not inconceivable, but it is pretty far-fetched. It’s generally true that biology ingeniously exploits most properties of nature to get things done, including electromagnetism, and we likely still have things to learn about that. But whatever those systems might be, it’s super unlikely that they have any meaningful interaction with a slightly salty bath, let alone one that’s relevant to aches and pains. It’s even less likely that any such effect wouldn’t be much more obvious in, say, sea water. Even if salty baths just bestowed a vague feeling of well-being and vitality, like mountain air, that would be biologically remarkable … but still well short of a useful medical effect. And in fact salty baths do not have an obvious mountain-air like goodness.
The Epsom salts article has now been updated with this odd tidbit.