Say you have been working with a client for over 4 years, you are there 2 nights and 2 days a week you have other clients on the other days.

They can do that if there's no conflict of interest with the agency who employs you now.

The family can always hire you privately too.

Depends upon the contract with the company that the family is filing. They may be agreeing to use only this company in their arrangements.

In caring for my Aunts in FL we had both family caregivers and agency aids during the week. I don't understand a company that would resent this arrangement -- it's none of their business. But the agency may have scheduling issues with its employees, trying to fill the time slots for your client's needs... due to the labor shortage the best aids often want the most hours on regular days so maybe this is their issue. Nonetheless, the agency can't demand anything from a non-employee -- if I'm interpreting your question correctly. If not, can you please clarify?

Unless the Agency is being hired for round the clock care I can't see what business it is of theirs what their clients do to fill in the hours they are not there, in my area with a high level of staffing shortages in caregiving professions it's become the norm for outside agencies to fill gaps even in long term care facilities. Maybe a different agency will be less demanding, and maybe it would be best for the client to stick to arranging for the amount of care they want and saying nothing about who or when they bring in other help.

Can who do what? Can the company insist that you become their employee, do you mean? No; but they can refuse to enter into a contract with your client if they're unhappy about working with you unless you are employed by them.

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.