After a lot of digging, I found out that Gulen came to the US during the Clinton administration.
A few words about Turkey, and Kemal Ataturk who founded modern Turkey. During WW1, Kemal Ataturk was the hero of Galipoli, although the Turks were on the losing (German) side during WW1 and were stripped of their empire, which was divided between Britain and France according to the terms of the secret Sykes-Picot Treaty (secret until the Bolsheviks released all those secret treaties after they seized power in Russia). Turkey was occupied by the allies, and Ataturk led a successful rebellion against the occupation in the early 1920s, managing to continue the genocide against the Armenians/Assyrians/Pontic Greeks in the process. Ataturk established the modern Turkish state and launched a series of political, economic, and cultural reforms aimed at transforming the religiously-based Ottoman Caliphate into a modern, secular, and democratic nation-state. I remember being taught that he got rid of the Fez (those little red hats with the tassel on top).
In the late 1990s, the charge against Gulen was that he wanted to undo secular Kemalism and return to the religiously-based Caliphate which ended in the aftermath of WW1. If that sounds familiar, the same charge has been leveled at Erdogan.
With those charges hanging over him, Gulen fled Turkey for the US during the Clinton administration. The charges went away in the early 2000s when Gulen's then friend Erdogan became prime minister. Gulen appears to have originally come to the US for medical treatment, or at least that was the reason he was admitted. He has diabetes, which of course is a rare and complicated disease for which he could only be treated in the US. The suspicion is that was a pretext and that once inside the US, he had no intention of ever leaving.
Somehow Gulen got a green card, despite the opposition of the Bush 2 administration.
The controversy was that he applied for an EB1 green card
(EB=employment based) as someone with extraordinary ability in education. Homeland and State both opposed the application, because whatever else Gulen might be, there was
"...overwhelming evidence that plaintiff is not an expert in the field of education, is not an educator, and is certainly not one of a small percentage of experts in the field of education who have risen to the very top of that field. Further, the record contains overwhelming evidence that plaintiff is primarily the leader of a large and influential religious and political movement with immense commercial holdings.”
In State's objection
"Because of the large amount of money that Gülen’s movement uses to finance his projects, there are claims that he has secret agreements with Saudi Arabia, Iran, and Turkic governments. There are suspicions that the CIA is a co-payer in financing these projects.”
And indeed, while Homeland and State opposed the application, 3 people described as CIA operatives supported it.