Many of my Facebook friends are very opposed to our nation's leader, and they make many strong comments against him. When I read Romans 13, it says to be in subjection to governing authorities because they were chosen by God for good, but how can that be true when a leader is evil?
As you quoted, Romans 13 teaches that all believers are to be subject to ruling authorities:
Paul explains that all who are in authority are there by God’s choice, and therefore when we refuse to be in subjection to governing authorities, we oppose God. Notice that Paul does not say that those who occupy positions of authority are always good people. Rather, Paul says that God is working through human government for good purpose, and sometimes God appoints evil men to occupy positions of authority because God intends to use their sin to accomplish good purposes.
For example, Nebuchadnezzar was the king of Babylon, and though he was not a “good” man, nevertheless the Lord used him to good purposes in bringing him to power. The king fought against Israel as God intended, taking them captive and holding the nation as slaves for 70 years. This outcome was good in the way it supported the Lord’s purposes by preparing the prophet Daniel and punishing Israel for sins under the covenant. Remember, good is a standard based on God’s perspective, not our earthly perspective.
So the ruler of any country is installed by God for His good purposes, and therefore we must remain in subjection to all authorities. Nevertheless, we may still oppose the policies of a government or object to a leader's judgments so long as we do so using legal means.
In Babylon, no mechanisms existed for opposing the king's decisions. He was the absolute authority in the land, so godly men and women had no option but to submit to his rulings. Today, we enjoy many rights for petitioning government and electing new officials, and we may take advantage of these lawful means as we desire without violating Romans 13. When we work within the rules of government, we are remaining in subjection to it as Paul commanded.
On the other hand, we must be careful to ensure our words and heart attitudes also comport with the Bible's teaching in Romans 13. While we are not required to agree with the rulings of a leader, we may not speak or act in ways inconsistent with subjection to the government. For example, we should guard our speech to avoid insults, slander, gossip, and the like when speaking of leaders. Similarly, we must maintain a proper and respectful attitude concerning our leaders since they rule by God's choice.
Finally, if a government's demands upon us run contrary to God’s word, then we must obey God while gladly suffering whatever penalty the government assesses for our disobedience. For example, Daniel’s friends refused to worship false gods in Babylon, so the king ordered them thrown in the furnace. They submitted to this judgment without opposition. Likewise, if our government demands we violate our conscience, we must decline to obey and suffer the consequences gladly.