Can everything we need to say about how to live be translated into rights talk? Possibly. But.
Consider an easy case. I can translate the statement that parents have a duty to care for their children into the idea that children have a right to their parents’ care. Not bad.
Now consider a harder case. I can translate the statement that we have a duty to seek the truth about God into the idea that we have a right to seek that truth, provided that we do not pursue it in ways that hinder the rights of those around us.
Something is lost here. Surely our duty to seek the truth is based on our need for the truth. But as rational beings we need not only to seek it, but to find it, and not only to find it, but to order our lives in by it. Both of these things require the assistance of other people. Rights talk doesn’t have much room in it for the idea of a community of persons who order their shared life by the truth.
Besides, which language we use has other ramifications. However firmly rights may be grounded in what is objectively just, they may seem to be subjective simply because they belong to individuals and need a subject to make a claim for them. Of course, the same might be said of duties, yet, psychologically, a difference exists. “My” duties in this sense direct my attention outward, to the persons toward whom I owe them. By contrast, “my” rights direct my attention inward, toward myself. This difference makes it very easy to view rights as though they were not really about objective moral realities, but “all about me” -- about the sheer assertion of my radically sovereign will.
The upshot is that if rights talk is the only moral language that we know, there is going to be a lot of moral slippage.