Is it permissible for an employee to register his wife and daughter in two health insurance systems that are not permissible by Sharia, if the employer automatically registers him in them?

All types of commercial insurance are impermissible, including commercial health insurance. It is not permissible for a Muslim to willingly enter into such forbidden contracts. So, if he has a choice to register his wife and daughter or not to register them, he is not permitted to do so.