“Between A Rock And A Hard Place” originated in the United States. The earliest known printed reference is Dialect Notes V (1921), which reads, "To be between a rock and a hard place, ...to be bankrupt. Common in Arizona in recent panics; sporadic in California."

The 'recent panics' referred to in the quote are the events surrounding the Bisbee deportations of 1917. In Bisbee, Arizona, in the early years of the 20th century, a dispute between copper mining companies and mineworkers developed.

In 1917, the workers, some of whom had organised in labour unions, approached the company management with a list of demands for better pay and conditions. These were refused and subsequently many workers at the Bisbee mines were forcibly deported to New Mexico.

It's logical therefore to surmise, given that the mineworkers were faced with a choice between harsh and underpaid work at the rock-face on the one hand and unemployment and poverty on the other, that this a probable source of the phrase.