I like that question. It recalls to me a conversation I had with my mother about a similar topic.

When St. Paul told his churches to obey authority, he was referring to the Roman Empire. When combined with other Scriptural references related to this, I see that three conclusions may be drawn:

1. We must obey authority as being placed there by God.
2. When there is a conflict of choices, we must obey the Law of God even if it is against the law of the State.
3. If the State simply allows for disobedience to God, this is not the same as commanding disobedience to Him.

As for a nation that no longer abides by its constitutional confines, we can and should use legal means to try to correct the imbalance. When it crosses the line into commanding disobedience to God, then we are bound to not obey those evil laws, even though it may be civil punishment.

That's my take on this.

  Suppose th consitution defines a ruler like this:
"He must be a man of politics, at least 35 years of age, with (naming other necessary qualities)."
  Now if under some social pressure by activists or different sects someone lacking in necessary qualities is elected, then if you accept him/her you are disobeying the law.
  The best example is if a 20 yr old man or a woman is elected.