Your question is an old one. First century Christians struggled with how much allegiance (if any) they were to give to Rome. The founders of this country struggled with it as well.
At one extreme you have the Jehovah's Witnesses (who do not pledge because they believe it is idolatry). At the other extreme you have Christians with a mindset of "my country right or wrong." A proper biblical response is somewhere in the middle.
Romans 13 tells us to obey those in authority. Other passages allow for civil disobedience (Acts 5:29, Daniel, etc.). Christians who live in a country with a godly government shouldn't have much concern about a pledge of allegiance. However, Christians who lived in, say Nazi Germany, might legitimately have reservations about a pledge of allegiance in that country.
I believe that if a Christian feels that it would be wrong for him or her to pledge allegiance, then I believe he or she should refrain. But if Christians then concludes it is wrong for every other Christian to do so, they are mandating a standard of behavior that I do not believe can be found in Scripture. Obviously Jesus Christ deserves our total allegiance, but I don't believe that a pledge of allegiance to a country undermines that.
Even though this issue doesn't necessarily involve the issue of civil disobedience, you might want to look at my transcript on the topic at the Probe web page (www.probe.org) as well as some of my other writings on Christians and government.
Thanks for writing. I hope this helps.