The mind-control technology being used by the Fatal Five was designed to overcome the conscious control of the mind of its victims. The subjects of the mind control would do whatever was required of them and this would not interfere with the use of their powers. Indeed the technology was designed to utilize the Legion as living weapons against the Science Police and the United Planets defenses.
In the episode of Justice League Unlimited, FAR FROM HOME (Season 3, episode 10) the Legion of Superheroes is under assault by the Fatal Five, a group of the Legion's most dangerous enemies comprised of Tharok, Validus, The Persuader, The Emerald Empress and Mano.
The Emerald Empress, Validus, The Persuader (w/axe) Tharok (half-cyborg), and Mano, with the Emerald Eye hovering menacingly overhead.
They bring three members of the Justice League (Supergirl, Green Arrow, Green Lantern) to help them rescue the rest of the Legion. Before they can begin a rescue, John Stewart is captured and place under mind-control by the Emerald Empress.
The alien mind-control device was part of a plan of the Fatal Five to use the Legion to attack the United Planets headquarters on Earth. Since this technology was created in the 30th century, such precise mind control technology has been used by various alien species. The origins of this particular device were not made known during the story.
The technology was designed to co-opt the free will of the metahumans without preventing them from using their powers, which they used to good effect in the final battle against Supergirl.
The technology would not stop a Green Lantern from using his ring because controlling his mind is not the same as preventing him from accessing his will. The technology simply prevents him from taking an active role in his actions. He could be given a set of instructions such as "stop Supergirl" and would use every means at his disposal.
This does not mean he was not fighting the device. It is very likely HE WAS FIGHTING THE MIND CONTROL since he used only physical attacks against her. These would have the least chance of harming her and he would know this. He could have just as easily created the radiation signature of kryptonite and killed her in a matter of minutes. Even without using the full power of the ring, Supergirl is eventually overcome and defeated by the combined power of the Legion.