For some reason "out-of-the-box" the Wordpress JSON API does not work on Nginx. I have tried several redirect schemes in the nginx conf. The only thing I have gotten to work is ?json. However, this does not work for authentication and registration.
As an FYI, I am developing a cordova application and attempting to use the WP JSON API for WP backend.
I found the solution to the problem. Make sure that permalinks are working properly before you assume (like I did) that it is an issue with the plugin.
I was able to correct permalinks for a wordpress site in a subdirectory on an nginx site. This article will help you if you face the same issue here