No, homeless people usually don't want a life without property; at least not the sane ones who aren't on the run from the law. Homeless people do have property of some sort; they push around a shopping cart full of their items and they usually have "spots" where they sleep at night. Even under bridges, tunnels, and underground homeless areas; every one of them has a little corridor to himself.

P.S. I am deliberately ignoring the much large population of working, "normal", homeless people. See "Persuit of Happyness".