No. There are not any U.S. federal or state laws that require a stockholder or LLC member be a U.S. citizen or permanent resident to form a U.S. company. Non-U.S. nationals can own all the shares of a U.S. corporation or be the sole members of a U.S. LLC. Nor must a member of the corporation’s Board of Directors or corporate officers own any shares (like “directors’ qualifying shares”). Similarly, all the members of the U.S. corporation’s Board of Directors and all its officers can, if so desired, be non-U.S. nationals and U.S. non-residents.
Most states require that all companies formed in the state have a registered agent in the state (more about registered agents can be found here).