Do you or your church support selective conscientious objection and conscientious objection?
Registering for the draft is a government law that does not violate any teaching of God. "Let every soul be subject to the governing authorities. For there is no authority except from God, and the authorities that exist are appointed by God. Therefore whoever resists the authority resists the ordinance of God, and those who resist will bring judgment on themselves" (Romans 13:1-2).
If you have objections to serving in the military, then don't go against your conscience. "Do you have faith? Have it to yourself before God. Happy is he who does not condemn himself in what he approves. But he who doubts is condemned if he eats, because he does not eat from faith; for whatever is not from faith is sin" (Romans 14:22-23). Fortunately many governments respect an individual's beliefs in regards to military service. But at the same time, the Bible does not teach that military service is wrong. See: Should a Christian serve in the military?