My wife and I have talked long and hard about this topic in the church and elsewhere.
From my research, Christians have come down on two fronts historically. One, that we should honor our country by rightfully submitting to government. The other is a more patriotic and nationalistic view. There are good people on both sides of the debate though historically it seems the vast majority of churches would not have a flag in the church (some exceptions exist with Episcopals, Catholics, and more state centered denominations).
To me, from my reading of history the more nationalistic and patriotic the church the more likely there is sycretism within the church. We saw this during and after the Revolutionary War where Christians embraced philosophies and theologies that were more nationalistic than Biblical. This led to implanting of variants of common sense moral reasoning which caused a great decline in Biblical theology of the day.
My wife and I have discussed whether a Christian should even pledge allegiance to any entity besides God and His Church. While we are called to be model citizens and submit to our nation's laws, do we really want to pledge allegiance to a fallen world system? Or, do we want to respect the laws of our land and pledge our sole allegiance to God?
In our homeschool, we began with having our kids pledge allegiance to the flag years ago. Since then we have ceased this and have elected to focus more heavily on Biblical teaching and training. Are we unconditionally dedicated to a Republic? I cannot say that I am. I am unconditionally dedicated to Jesus, His Word, and His Church. I love my country and will follow her laws insofar as they agree with the Bible. Yet, my allegiance is never with America, it is with God alone.