Work still has to be genuine and effective as with any EU national.
If a person has gone straight from working to ESA(c) and are temporarily unable to work due to injury or illness they will be ok to continue claiming as retaining worker status. If it is due to a permanent injury / illness there are additional criteria that need to be met.
With JSA a person who receives JSA(C) can qualify for HB and CTB but as with the ESA(c) they need to retain worker status.
ESA(IR) and JSA(IB) passport a person into eligibility without you having to complete any checks on EU nationals status.