Yes, he should he had no right to leak the information he did. He put a lot of our lives at risk by leaking highly classified information. He is running because he is a chicken and he knows if you do the crime you should more than likely do the time.
America has a good tradition of civil disobedience used to make a point and help the nation grow and progress. However, most of those who are successful in civil disobedience are committed enough to their point to stand their ground and do their time so they get their day in court. Bashing your country and leaking its secrets while on foreign soil is not civil disobedience, it is treason.
I think that any country that currently has or is harboring Snowden should extradite him back to the United States of America. Snowden is a criminal and enemy of the USA. And if a country want to have a good international relation with the USA, they should send Snowden back to America.
I do not believe Snowden should be extradited to the United States. If he were to return to the United States he would more than likely be jailed and I would say it is possible that he could be harmed by the government. I believe Russia should continue to offer him asylum.
While Snowden likely did break the law I think he was justified in doing so since the documents he leaked sparked a massive debate in this nation over the nations intelligence force spying on the general public, I think Snowden would be killed if extradited back to the United States.