Can my POA force me to live with them when I want to go home?

Are you the POA, or are you an elder whose child or other loved one wants you to come live with them ? Certainly you can make your own decision in either situation, but you also need for this not to become a tug of war. Are there other choices for caregiving aside from one or the other moving?

No. Most POAs do not live with their parents. Many live in totally different states.

No, a POA does not have that kind of authority.