Yes, families should be required to take care of relatives with Huntington's disease mental impairments, because it is much better than the people being left to the state. A person is obligated to the well being of their relatives, even if the relatives have physical or mental disabilities. Families need to care for their own, rather than the government.
Family bonds are important in the best of circumstances, but in our modern society especially in the United States they are very fragile. People live at a distance or they may have not been treated well by their relatives, or any number of things might have happened so that no care should be enforced ever.
Families should definitely help relatives with Huntington's, but they should not be required to do more than they can. The healthcare system should pick up the slack and help anyone with the disease who is unable to get help from family members or other sources. Forcing families to help when unable just causes instability and difficulties for the family.
Since the proper care and treatment of a person with Huntington's Disease is both time and labor intensive it would unfair to saddle a family with no choice but to surrender their life and assets to care for this person. While many families would make the choice to care for their loved one, it should be a choice and not mandated.
I do not believe this question is specific enough to give an ideal answer. Huntington's Disease affects each individual differently and on-set and progression can vary widely. Some may be able to take care of themselves for a number of years while others may require medical assistance as well as assistance with daily life activities. Family members, when referring to parents, should of course be expected to care for these individuals for as long as possible for their circumstances. Other relatives, not so much.