It may depend a little on how serious the back problems are, but assuming they are only temporary he's still entitled to claim Jobseekers' Allowance -- as long as he is seeking employment, that is, with more information available at the Jobcentre. That he left his previous job of his own choice is neither here nor there, and he should be encouraged to claim for that -- or, if possible in his area, Universal Credit (see link at the end of this answer).
Your sister can also consider applying for Universal Credit in the same way -- again, only if the couple lives near a jobcentre listed below. Of not, then Working Tax Credit may be available but the conditions for your sister are that she must work at least 30 hours per week and, otherwise, she should be able to claim regardless of income.
Just to be irritating, the above changes depending on whether or not you have any children (still living with you). But either way, encourage both to apply for benefits -- JSA for your sister's husband, and Working Tax Credit for your sister, or Universal Credit for both of them if possible.
https://www.gov.uk/guidance/jobcentres-where-you-can-claim-universal-credit