Or are we to presume that only those who believe in God have rights?
I will disagree, though I see your point.
"Rights" aren't given or taken, they just "are". Now in order to preserve or obtain or obey the "natural law" and it's responsibilities you will need to "fight" for them. Because others ,due to envy, greed, or other social pathologies will want to take what is not theirs in property, money or labor .... by removing it from your possesion against your will or desire.
In other words bad people want to steal what they don't want to work for...... most politicians, lawyers, environmentalists and other assorted "liberals" fall into this category.
As far as "God" getting into the picture...... well, everything we have, will have and hope to have is given to us by him. That is, the only things worth having, hoping to have, or will have is through him.
If you're up for a suggestion, I'd recomend CS Lewis's "Mere Christianity" and "Screwtape Letters".... he kind of sums it all up in a tidy little package from a former atheist's and university scholar's point of view.
The rights you describe are the rights pertaining to particular cases. Think more abstractly.
The universal rights of individuals are indeed granted by God. For example, the right to opportunity, the right to choose good over evil.