In the US, if I paid someone to help simulate my new product design using some computer software (but the idea is completely mine), does he have any right in the design / patent?
They have already signed an NDA for me, if that matters.
No, even if he hadn't signed an NDA. No one is entitled to be on a patent unless they have contributed materially to the basic claims and novelty of the invention. If they end up adding something useful to the invention, then you should add them. If you don't, they could make a legal argument in the future that they should have been, especially if money is involved.