Joe Cirincione makes the easonable case that the USA could not successfully shoot down the North Korean ICBMs. Joe Cirincione is president of Ploughshares Fund and the author of Nuclear Nightmares: Securing the World Before It Is Too Late.
The North Korean missile was about 770 kilometers (475 miles) over Japan at the apogee of its flight path. None of the theater ballistic missile defense weapons in existence can reach that high. It is hundreds of kilometers too high for the Aegis interceptors deployed on Navy ships off Japan. Even higher for the THAAD systems in South Korea and Guam. Way too high for the Patriot systems in Japan, which engage largely within the atmosphere.
Anti-missiles are currently designed to intercept in the the post-mid-course or terminal phase, when it is on its way down, coming more or less straight at the defending system.
Patriot is meant to protect relatively small areas such as ports or air bases; THAAD defends a larger area; the advanced Aegis system theoretically could defend thousands of square kilometers.
There is almost no chance of hitting a North Korean missile on its way up unless an Aegis ship was deployed very close to the launch point, perhaps in North Korean waters. Even then, it would have to chase the missile, a race it is unlikely to win. In the only one or two minutes of warning time any system would have, the probability of a successful engagement drops close to zero.
What would work to intercept North Korean missiles ?
If the US chose to weaponize space (violating current treaties), then anti-missiles or railguns or other weapons could be placed on satellite(s) positioned over North Korea. Those systems would be firing down at the North Korean missiles that were boosting upwards.
The other option is to spot the missiles as they are being readied to launch and having systems to rapidly strike and destroy the missiles before they launch.
Another option would be to find the missiles where they were being stored and destroying or sabotaging them.
Google Loons are now able to station keep in place in the stratosphere. A large militarized google loon could carry anti-missile systems as well to shoot down at a target.