Religion does affect government structures. If a person is told to go to war, vote or salute a flag and does not, it's not out of disrespect. It is because those actions themselves disrespect God's views on the matter.
Notice in Matthew 22: 17-21 and see in Jesus' day when the scribes and Pharisees cunningly asked him " Tell us then, is it lawful to pay head tax to Caesar or not?" Jesus had asked for a denarius which was a tax coin. He asked them " Who's image and inscription is this?" To which they replied " Caesar's." Jesus then said " Pay back, therefore, Caesar's things to Caesar. But God's things to God."
So while we are loyal to pay taxes and respect government and kingly authorities, if they ask us to do something scripturally unlawful, we respectfully decline and if they persecute us for our choice, we let them.