Natural law is law based upon nature as opposed to positive law which is based upon human decree. Natural law is supported by what works as opposed to what he want to work, e.g. genetic inheritance (natural) versus lysenkoism (positive). Progressives supported the latter but the former won because it was consistent with nature.
An adherence to natural law means you are willing to change as you learn more.
The Bill of Rights doesn't contemplate our rights, it recognizes them. The question is never where do we get our rights but where does the state get its powers. Progressives see the state as bearing collective power by default. Conservatives and libertarians, including atheists, believe the state has only what powers we give.