Through the years I've given this issue a great deal of thought. I used to be a gung-ho patriot of the United States and its military. However, as I look in the bible I don't see any calls upon Christians to take part in any form of violence on the part of a nation state. Instead, we're expected to be peacemakers in the name of Jesus Christ. As Christ-followers, we may suffer and even die for the cause of the gospel. That's our real mission.
God is the author of life. He creates it. Because of that, I have no right to take it. My citizenship is in heaven (Philippians 3:20). Only secondarily am I a citizen of the USA. I love the kingdom of God, but I do not love the USA. My service is to my Lord and Savior, not to a secular state.
The United States is not a new Israel. Its soldiers do not fight for God. Rather, they fight for political freedom and the American way. The problem is that God has not called upon his followers to kill-or-be-killed for political reasons. He is much more concerned with freedom from the bondage from sin than freedom from political oppression.
I cannot imagine joining the military, training for combat, and then being faced with killing someone. It flies in the face of what Jesus has called us to do. Our Lord instructs us to lovingly serve others to the point of dying for them. Why in the world would I take lives just to supposedly advance the cause of the USA?
I do not stand in judgment over Christians who have joined the military. This is certainly a difficult issue. Additionally, I suppose that it is possible to be part of the military without taking part in killing (chaplaincy for example).
As for defending my family from invaders (foreign or domestic), that is a different issue. In that case, I would not hesitate to kill someone who is threatening to kill my family. As head of my household I believe God demands that I defend them (not that I would take any pleasure or satisfaction in it). God does not, however, expect me to defend a secular political state.
On Memorial Day I always have mixed emotions. I'm glad to live in a country where I have semi-religious freedom. At some level I'm thankful to those who died defending that freedom. Despite this, on principle I don't support the killing of other human beings for political reasons.
The cross is not wrapped in the American flag.
Jesus Christ could have come to earth as a political conqueror to throw off Roman occupation of Israel. However, he had a bigger and more important mission to accomplish. He gave his life as the gospel. Those of us who call Jesus Lord have been commissioned by him to take this gospel to the ends of the earth. Like many others before us, we may be called upon to die in doing so.
The Great commission is too critical to waste our time defending our country's political aims via the military.
Ultimately, it is because the gospel is so important that I could never serve in the military.