They are still citizens. There is a contract between the service member and the government. The service member does not lose their rights or citizenship or any of that, but they do agree in contract to do certain things such as following the legal orders of those appointed above them, and adhering to the Uniform Code of Military Justice (an additional set of laws for the military members).
The religious exemption is suspect also. Kind of hard to obey "Thou shall not kill" if you are in the military.