Natural or universal rights does not require theism. Robert Nozick is famous proponent of the secular based position that property is a natural right.
Nozick's position on the existence natural rights is simply not grounded. He appeals to intuition and to the reader's morality to appeal for their existence, but he doesn't (and cannot) actually deduce their existence once he forgoes theism. He makes a few appeals to Kant, but they obviously cannot be sufficient, Kant's conditions are merely necessary. I'm very confused by your reference to Nozick on a discussion about the grounding of natural rights when Nozick himself admits that he cannot justify them - he simply assumes Locke, which himself uses a theistic argument, in ASU. If you want, I can get the quote, but I don't have time to skim it until I'm home from work.
At the end of the day, secular natural rights is an intuitive and appealing but ungrounded position that cannot be logically justified, hence why it is threatened by theism. It is no wonder that positive rights and social right theory only really emerged after the Renaissance.
Hume's original text describing the is-ought problem is specifically targeting justification of ethics on god. Laws that no one is ought to abide by are no laws but nonsense.
Is murder immoral because god hate murder, or do god hate murder because it is immoral?
> At the end of the day, secular natural rights is an intuitive and appealing but ungrounded position that cannot be logically justified
Many will argue that no moral theory can be logically justified, and that the search for logical justification is category error
> Is murder immoral because god hate murder, or do god hate murder because it is immoral?
We agree here, but that isn't how religion solves the problem. Religious laws are also enforced by threats in the afterlife and violence in the present life, not merely by reason, so they do not need to solve the is-ought problem like secular laws do. Of course, religious law has other problems.
> Many will argue that no moral theory can be logically justified, and that the search for logical justification is category error
I agree completely, hence why it is impossible to derive rights that are logically justified without an appeal to God. The comment I was replying to claimed there were logically justified rights which have to follow from logically justified moral theories unless they are decreed from beyond reason, and since the latter is a category error, so is the former.
