My cousin is 16 she lives in the USA illegally. She would really like to work for the FBI someday. If she leaves the country before she is 18 ( I understand no bar will be given), and somehow manages to get a citizenship of the US, will the fact that she was once here illegally prevent her from ever getting to work for the FBI or the government? Also if that's the case, is it final or is there another way to hopefully work with the Government?

I don't think her chances are great of either working for the FBI or obtaining citizenship.

To work for the FBI will take an advanced education, and often either police or military employment first. (or to be an attorney) There are perhaps 1000 people apply for any one person hired. So yes, most likely unless she has some very special skills, in languages or other fields, the chance of working for the FBI is slim.

There are 100's of thousands of different types of government jobs, so yes if she goes out, and comes back legal, there is a chance. Also if she would come back legal, go to law school and ace it, there are always exceptions. But at that point, if she aced a law school, there would be more opportunity in private industry.