Alrighty, so this is going to take a lot of finagling and essentially what could be called "rules-lawyering" 

The first thing we need to determine is could Sauron somehow connect himself with all of the Lantern Corps? Obviously, we can't have him being responsible for the formation of the rings because ...
 well then obviously, yes he could have. We're better than that.

So, how could Sauron sneak his way in? Well, this is where things get messy, and rely on blending the two universes in a semblance of fairness.

A power ring manipulates a portion of the emotional spectrum, which helps us to get started on the path to universal domination - to hijack the green lanterns, Sauron will need some innate connection to Willpower. Looking through that wikipedia entry, the part that caught my eye was "Entity" which redirects us to:
https://en.m.wikipedia.org/wiki/Ion_(DC_Comics)
So basically, Ion is the entity, or spirit, of willpower. It essentially powers all of the green lantern rings as the universal collective willpower. I have not checked beyond the yellow power rings where I found mention of a Parallax entity (collective universal fear) but I would assume that every ring possesses an entity, (nekron further reinforces this.) 

The closest approximation to these entities in LOTR are the Ainur (primordial spirits.) Of which Melkor - Sauron's boss - was unequivocally considered to be the most powerful, and possessed aspects of all other Ainur. This is our in. Melkor in our shared world shares the embodiment of Will, Fear, Life, etc with these other beings such as Parallax, Ion, and Nekron. Sauron, being a Valar, and likewise, the strongest of the bunch, was merely a step beneath them. We can assume that Melkor granted certain boons to Sauron, the exact nature of which I couldnt determine, but if we equate Sauron as an avatar to the Ainur whom he serves, we can grant him Valar status of each Primordial Spirit (entity).  Sauron was initially bound to the spirit of invention or something (I suppose Hephaestus is a good IRL parallel, but its nice to know that LOTR has a bit of silliness as well) which is why he chose rings, and was the Lord of the Rings, which gives him a second in, since the rings are clearly an invention. Moving on.

As we all know, Sauron imbued part of himself into the one ring, this, combined with the other rings he aided in crafting allowed him to spread his influence... on second thought I'll just put this right here:
One Ring to rule them all,
 One Ring to find them, 
One Ring to bring them all, and in the darkness bind them

Breaking this down, One Ring to rule them all: This is the core of our argument, we simply have to extend this to the power rings. 
One ring to find them: In other words, one ring to connect him, to all the rings (of power) he had a hand in creating.
One ring to bring them all, and in the darkness bind them: this tells us that through the one ring, Sauron was able to reach out and spread his darkness to those who wore his rings and finally bring them under his control.

The obvious counter to this would be, "See, Sauron needed to have a hand in making these rings to put his touch upon it" This is true. In the world of LOTR where rings of power are rare and far between. In our mixed world however we have literal legions of people with Rings of Power that all are sharing/channeling atleast a tiny fragment of Melkor, and therefore Sauron.

In otherwords, Sauron doesn't need to craft a new ring to subjugate the Lantern Corps. He already did.

[Img]https://i.kym-cdn.com/entries/icons/facebook/000/013/034/yeahsciencebitch.jpg[/img]

(As for Nekron, Ion, and the others? They would likely be safe. Under the above presumptions however, I doubt even the white lanterns would be untouched)