Since there is no draft, why would there be a need for positions for conscientious objectors? They're all voluntary. Nobody forced them.
If someone becomes a conscientious objector during his time in the military, there should be a way to either assign them to non-combat duties or discharging them. However, IIRC, conscientious objector status means that you do not agree with any war. This guy seems to dislike the current war based on political, rather than moral, views.