America has always been considered a "melting pot" - a place where different people from all over the World can come to and live together as a cohesive group. Muslims in the United States have always integrated fairly well into American society. Therefore, the United States should not do anything to antagonize the American Muslim population. In short, no, the United States should not follow Japan's example of putting Muslims under surveillance.
Muslim is a religion and a religion can't be used to categorize people's actions. Only a small proportion of Muslims can be viewed as terrorists so putting the entire religion under surveillance is just wrong. Now the United States should put certain Muslims under surveillance but shouldn't follow Japan a put them all under surveillance.
There is no need to put extra surveillance on Muslims. They have a right to their protection and deserve to have the same level of surveillance as a Catholic or a Jew. All Muslims are not radical, so the countries should be putting more surveillance on those whom might be radicalized.
The overwhelmingly vast majority of Muslims are not terrorists, nor do they want to become terrorists, so putting surveillance on them would be a huge waste of time, resources, and energy. Just because a few violent people in a group commit terrible crimes, it does not follow that everyone in that group is a potential danger to society. By that logic, it would actually make more sense to put surveillance on every single young white male in the United States, as they perpetrate so many mass shootings.