The American military only classifies one as a conscientious objector if one is opposed to all war for any reason, and only on religious grounds. In other words, if you believe a particular war is immoral, that doesn't count. If you would fight in a different war somewhere else or over a different reason, you would not be considered a conscientious objector. If you objected to a specific war or any war on any grounds except a complete religious objection to all war, you would not be considered a conscientious objector. If you were willing serve in a non-combat role, such as a support unit or as a clerk, or even as a medic, you might be able to do that. On the other hand, many documented cases exist where draft boards or military authorities have denied such requests.
False. The person can refuse to go to war also for religious reasons too and still be considered a conscientious objector. Also the person can participate in the war through non-combative means and still be against the war itself.