That is definitely letting the family go too far. Legally, no. Unless they're paying for private duty nursing 24 hours/day, 7 days a week they do not have any rights over what nurse takes care of their family member.
Refusing to take the patient can put your job at risk as there are few protections available unfortunately.
I'd get a consensus for how the other nurses "on the list" feel. In facilities that I work in, you get who you get unless complaints are filed against this nurse or that.
It's definitely a difficult situation for you to be in. I'm sorry to hear that your hospital administration has caved in like they have. Doesn't say much for their support of their own nursing staff.
Andrew Lopez, RN
http://www.4nursing.com