You have a right to life, you are not entitled to be taken care of, these are entirely different things. He's not saying you owe anyone anything. Basic life sustaining resources require labor to be made useful, shelter doesn't just appear out of nowhere, someone has to labor to make it: you do not have a right to someone else's labor, you must do your own labor to take care of yourself, it is not owed to you. Being required to labor for yourself does not imply you owe anyone anything.
> The idea of wealth you're referencing is also rooted in the current system.
It really isn't, wealth just means stuff. All systems will have it, it has nothing remotely to do with just the current system. Stuff doesn't just exist, someone has to create it, that someone by natural right owns what they create; no one is entitled to another mans efforts.