What do I list as my permanent country if I am a US citizen living abroad?

The question is not about citizenship, but rather about your residence. If you are temporarily living abroad (for example, in an exchange or study abroad program) you should select the United States as your permanent country of residence. 

If you and your family permanently live overseas, you should select the country where you currently live as your permanent address. However, it may be helpful to follow up with your schools to ensure they have clarity on your application.