1. Green card holders can’t get secret clearances
2. Naturalized citizens can get secret clearances. Most of naturalized citizens in US military have it. But if they couldn’t get it they have a waiver for it.
3. Doesn’t really matter whether you are naturalized or born in US; getting TS clearance is hard for all citizens. There are tons of people couldn’t get their TS clearance that they had to switch their jobs inside of military.