But what right does the U.S have to put restrictions on other countries for this?
Individually, we don't depending on what you mean by restrictions. We are part multiple defense packs with Japan, South Korea, Australia, New Zealand plus other agreements with many southeast Asian nations to ensure peace and stability. The authority to pressure NK to deter a nuclear program or other offensive military programs is granted by the collective nations who are most vulnerable to such military offenses among other international bodies.
That said, the restriction that we place upon NK are not direct but indirect. We cannot enter NK and stop their programs (outside of a declaration of war and authorized by the United Nations...preferably). But we can apply sanctions. Further, we work on the international level to impose sanctions that are approved by a sanctioning body such as the UN. There are about 2 dozen UN sanctions and an equal number individually from the US, China, Japan, South Korea, the EU and Australia.