I've been trying to find information on "or bust"

IANALinguist, but I do think "California or bust" was the main popularization. "The History of California Vol III" by Theodore H. Hittel from 1898 gives some backstory. He describes the arduous crossings of the Great Plains during the gold rush in 1849:

mothers might be seen wading through deep sands or climbing steep rocks and dragging along crying children; or sometimes fathers, strong when they started but now pale and emaciated with hunger and fatigue, carrying infants on their backs and appeasing their ravenous appetites from the carcasses of dead horses or mules [...]

But people were still willing to risk it all to get there:

The humors of the plains were usually of a rough and sometimes coarse nature. The large canvas-covered wagons were called "prairie schooners," and many of them carried mottoes such as "California or bust," "No return tickets by this line," or something equally as expressive but unrefined.

Etymonline says "bust" meaning "go bankrupt" is from 1834, 15 years prior to this, so I'm guessing that's the original meaning. I.e. "Reach California or lose everything: we won't give up until it's one or the other"

I don't know how accurate it is, but an explanation is offered here:

"California or bust" is the archetypical phrase, model for so many imitations. The Great Plains of the U.S. were struck, in the 1930's, with two catastrophes: drought and the Great Depression. So many dirt farmers were left with no dirt, just dust and debt, in the wind storms destroying their farms, that some of them headed west on Route 66 towards what they hoped would be greener pastures in California, the "Golden State". Some of them had signs "California or bust" on their shabby jalopies and trucks, loaded with their families and few possessions, as sometimes shown in newsreels. When they got there many were known as Okies, having started in Oklahoma. See Steinbeck's "The Grapes of Wrath" for more about the Okies and their trek. Not all of the Okies were lifted up once they reached California, and there are still pockets of people referred to as Okies by other Californians.

As for "bust," I suppose they meant they were going to bust their guts trying, and failure would be complete. At any rate, it was a popular slogan of the time, no longer in vogue. You could say, for instance, "No. 1 or bust." A similar phrase was "... or nothing," as in "complete success or nothing." Not being a popular historian I can't produce actual examples

If the explanation is to be believed, I'm not sure why the author concludes, " I suppose they meant they were going to bust their guts trying, and failure would be complete". It would seem to me that the meaning of "or bust" would refer to the possibility that the vehicles that those from the Great Plains were driving would break down on the way to California or their willingness to resettle wherever it did.

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Edit: Edited for clarity.

It goes back to at least the Gold Rush. "Pike's Peak or Bust" dates from the 1850s:

Almost every wagon bound for the gold mines has some motto or other in large letters on the cover. A few weeks ago we noticed one with : 'For Pike's Peak or Bust.' Yesterday the same wagon returned with this addition to the motto: 'Busted by G----d'

Pike's Peak is a location in Colorado where gold was found at this time.

Here's a mention of the phase "California or Bust" in "St. Nicholas: A Monthly Magazine for Boys and Girls," Volume 3, Part 1, 1876. Its meaning is not explained, though.

Well, as someone else mentioned bust is an American variant of burst, so it cant be older than the American colonies themselves.

I also like that the etymological semantics of bankrupt are reawakend in the idea of go bust/burst!!!!!

Blackjack. Aiming for the best/highest, "or bust"

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