Your client has 24 hr care and 2 of the caregivers have left . They are thinking about hiring a company to provide help. My client nor I want to change our arrangements, however the company will let me continue working there if I work under them... Can they do that?
The family can always hire you privately too.
I can see their point of view. They probably have not only agreed rates of pay, but also agreed ways of working, standards of practice, training, registration, vetting and so on, and having you on the premises 2/7 days could create problems.
They still can't dictate terms to your client, who is free to choose whatever care services s/he likes. Your client has to weigh up the pros and cons of "guaranteed" 24/7 provision versus possibly losing you, if you can't come to an arrangement everyone is happy with.
This gives you a lot to think about, assuming that you're currently self-employed and working as an independent caregiver. But I would find out about what terms exactly they're offering and think about it, rather than just say no.