2 Corinthians 10:3
For though we live in the world, we do not wage war as the world does.
Looks like you cherry-picked the NIV for this verse; other translations make it clear that Paul was talking about spiritual, not physical warfare:
For though we walk in the flesh, we are not waging war according to the flesh. (ESV)
For though we walk in the flesh, we do not war according to the flesh, (NASB)
For though we walk in the flesh, we do not wage warfare according to [what we are in the] flesh. (Yes, even the NWT).
Read the context. This verse has nothing to do with whether a Christian should serve in the military. It's talking about spiritual warfare against false teachings (notice verse 5).
In Luke 3:14, soldiers came to John the Baptist and asked what they should do (in response to the message he had preached). Did he tell them that their military service was unacceptable to God and that they should resign from the army? No:
Luk 3:14 Soldiers also asked him, "And we, what shall we do?" And he said to them, "Do not extort money from anyone by threats or by false accusation, and be content with your wages."
Similarly, there is no record that Cornelius was asked to resign as a Roman soldier (even an officer!) in order to be pleasing to God (Acts 10).
Paul used military illustrations constantly (Ephesians 6, for example, but there are many others). Why would he do that if military service were repugnant to God? Can you picture Paul using adultery or murder as an illustration of spiritual virtues? Of course not - but he used military imagery over and over again.
It's true that some historians report that the earliest Christians did not join the military. But why was that? Was it a moral position taken on biblical grounds? Or could it have been that the earliest Christians lived under the Roman empire, which spent a lot of time trying to hunt them down, destroy their scriptures, and throw them into arenas to be eaten by lions? If that was the business in which the army was engaged, one can easily see why the Christians would refrain from joining - not because military service in itself was forbidden, but because the military AT THAT TIME was their greatest enemy.
I see no biblical objection to military service in a nation where the military is not engaged in the persecution of Christians. I respect that other Christians may disagree with my position. What I don't respect is that an organization dictates matters of conscience for its members in a pharisaical fashion and requires parents to shun their children who make a conscientious decision that differs from the organizational position.