I have some experience with home health care, though not live-in. If the family goes out, they usually do so at the end of my shift, and I get released early because they usually have enough people to take care of what I would normally do. Otherwise, if I went with them I would expect to pay for my own meal because they are taking their family member out, not me. I would order something I could afford to pay for, even if it is only a small salad, and then later eat my brown bag lunch.
Edited to add: My duties usually include various types of housework as well, so I would definitely appreciate some time alone to do some of the chores that can be more difficult when taking care of someone. (For example: sometimes it's hard to get the floor mopped if the person you're taking care of has the habit of wandering through the kitchen at random times, because you don't want them to slip and fall.)