It doesn't seem an unreasonable place to start looking at all, especially as you're right - a lot of the surname usage seems to concentrate/originate in that area of Somerset. (Bear in mind, though, that any UK port might have an occasional ship going to a foreign port even if it was unusual, and that all British ports were closely connected by regular coastal traffic anyway, so it would be impossible to rule out all other locations for sure just on that basis.)
I also have two sets of Washer families from that area on my tree, and while one of my lines fades out before that, and the other is a sideline on another tree that I haven't been able to expand yet, my Washers also moved between the West Somerset coastal area, Bridgwater (another port) and Bristol. So it's possible I might be able to help give pointers with some of my own researches - although sadly I do not have a ready-made missing John to match your Boston runaway!
But if you want to send me a PM and maybe let me know a bit more, I'll see whether I can fish out anything useful for you. Good luck with your searching anyway! And while it's not the rarest name, it also isn't super common - so maybe even if we can never prove it, we might still be long lost cousins from a very long way back, who knows?

(Bear with me if I'm slow to respond at any point - I'm not always that well at the moment, but I will always still get back to you eventually.)