Dewey's objection to Natural Rights does not stem from a belief that the individual has no rights. It stems from a rejection of the concept of natural rights, which holds that there are universal, eternal, divinely ordained rights. He doesn't think Rights come from God, nor does he think they are absolute, timeless or unchanging.
He believes that Rights emerge democratically from free citizens who take an active role in constructing the world in which they live. Dewey believed that citizens should be empowered to vote on the standards by which they live - and select Rights which reflect those standards.
The reason he does not believe in one set of Absolute Universal Rights for all time is that he thinks, with Darwin, that humans evolve and change. So for instance, in the 1800s, a white man did not have the right to marry a black woman. However, over time, our moral and political standards changed - so too did our beliefs about human nature. We no longer think of blacks as less human (or fundamentally different) than Whites. And soÂ… as our beliefs changed, we revised our Rights according to our changing situation. Therefore, today, a black human has a Right to marry white Humans. Rights changed with our evolving beliefs/standards. This contradicts the theory of Natural Rights, which holds that there is one set of permanent unchanging Rights delivered by God (or Nature or History, in the Hegelian sense).
If you read Dewey's "Democracy and Education" you will see what he means. He believes that students are not mere receptacles for the traditions/beliefs of their ancestors - they are instead active participants in creating the kind of world they live in. He thinks that a citizenry should have the freedom to analyze their world and select their own personally meaningful values/traditions. For Dewey the most precious thing is human freedom, specifically the Right to choose the kind of world you want to live in. He didn't subscribed to anarchy, with each person deciding for himself which Rights/Laws have authority; he believed in Democracy, and returning power to the people by allowing them to debate and vote on the rules by which to live.
What really separates Dewey from the Right is that he had greater allegiance to Democracy than Tradition or Religion. For the Right, we must live in the world as God intends. For Dewey, we must live in the world as the people intend (through democratic institutions like public debate and voting)
There are great arguments against Dewey's concept of Rights, but it's false to say that his objection to Natural Rights is an objection of individual Rights per se. They are two very different things. This post reflects the kind of intellectual dishonesty and laziness that has marked the Right since it became an activist movement. Rather than a concern for truth, our greatest political party has become a collection of think tanks which spread talking points to useful idiots.