Christians may be conscientious objectors but not just to avoid military service or to escape the draft.
One must be totally and philosophically opposed to a particular war or to military service in principle and not just when it is convenient. There is also a moral obligation to support and defend your homeland.
So conscientious objectors can be exempt from combat or military service, yet still be patriotic, by rendering civil service in some other capacity (for example, volunteering for the Red Cross, hospital, Peace Corps, or engaging in some form of public or private charity work).
If someone enjoys the benefits and privileges of citizenship but does not pay taxes or is unwilling to defend the nation that protects him, that would be injustice. You can defend yourself, your homestead, or your nation, and still be a good Christian.
Saint John the Baptist told Roman soldiers to be fair and did not tell them they had to leave the army. “Soldiers also asked him, ‘And what is it that we should do?’ He told them, ‘Do not practice extortion, do not falsely accuse anyone, and be satisfied with your wages'” (Luke 3:14).
Jesus said “render to Caesar what is Caesar’s, but to God what is God’s” (Mark 12:17). From these two passages, the Church has historically seen no contradiction in Christians providing military service to their homeland.