Here's the Johnny Cash song, obtained from http://www.allcountry.de. It doesn't seem to have much of a connection to the Dylan song, or to John Hardy.
-Joe Offer-

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HARDIN WOULDN'T RUN
Words and music by Johnny Cash
As recorded by Johnny Cash on "Johnny Cash Sings the Ballads of the True West" (1965)

I know a man whose plow-handle hand is quicker than a light.
Wes Hardin is his name, they say; he travels in the night,
For he might have to kill or walk around a fight.

And if you ever saw Wes Hardin draw, you know he can skin his gun.
He won't say how many tried and died, up against the top hand,
Up against the wrong man, 'cause Hardin wouldn't run.

He rode in like a Texas wind, took the eastbound train,
Goin', goin' with Jane Bowen till the lawmen caught up.
"So long, Janie; chin up! I'll be back again."

Off he went to Huntsville Prison. "So long, Jane," he cried.
Fifteen years she waited till her heart broke and she died,
And she left that bad land to wait up in the sky.

Free at last, the payin' past for all the wrong he did,
First free air they let him breathe since he was a kid.
So let him come and let him go; let him deal and bid.

Near the border in El Paso, lawyer reads a sign.
You won't find him there for business every day at nine,
For business is real bad; one client's all he's had in quite a long long time.

Then sheriff Selman's boy broke into Wes's woman's place.
Up she jumped and pistol-whipped him, kicked him in the face,
And John Selman demands revenge for this disgrace.

You could see ever' night by candlelight in Hardin's favorite bar,
She'd be hangin' on his arm and very late they'd leave there,
Headed for the Goose Hair, glad it wasn't far.

Right through the swingin' doors, John Selman came with a blazin' gun,
Wes Hardin chug-a-luggin' redeye, got him in the back of the head
John Wesley Hardin fell dead; Hardin wouldn't run.

© 1962 Southwind Music