Canadian Citizen Living in the U.S. - Teaching in Korea

Help! I am a Canadian citizen, but have been living in California for the past 10 years. I am a U.S. permanent resident. However, I am planning to teach abroad in Korea for a year. I know I have to apply for a re-entry permit to come back to the states if I plan on leaving the states for over 1 year. My question is -- does this affect my U.S. citizenship eligibility?

Also, does anybody know about the criminal background check process for teaching in Korea? Would I have to get a background check from Canada or from the United States? Hmmm. If anybody has answers, please let me know!