Jayne's birthplace hasn't been revealed.
However, several pieces of dialogue from Jaynestown imply that he isn't from Canton.
Oh, listen, Mal -- I was in Canton a few
years back, I might have made me a few
Look, Mal, I got no ruttin' idea. I was
here a few years back, like I said. Pulled
a second-story, stole a lot of scratch from
the magistrate up on the hill. But things
went way South. I had to hightail it.
They don't...put you on a pedestal in town
square for that.
Uh...I'm no good with words. Don't...don't use
'em much, myself.
But I want to thank you all for being here, and
for thinking so much of me... Far as I see it,
you people been given the shortest end of the
stick ever been offered a human soul in this
crap-hill 'verse....But you took that end...
well, you took it. And that's... well, I guess
He also doesn't react when Wash insults the town. Knowing Jayne's general attitude, he probably would if he actually came from there.
We gotta go to the crappy town where I'm the hero!
In the episode, the townspeople are completely wrong about Jayne. They turn his random, unintentional actions into an act of purposeful altruism. The "his people" portion is likely them just exaggerating Jayne into a typical folk-hero.