No, you are not susceptible to a gaze attack through an Arcane Eye, regardless of distances.
The rules on gaze attacks state:
A gaze special attack takes effect when foes look at the attacking creature’s eyes.
Emphasis added, because the words “look at” are key. I consulted three dictionaries for the relevant definition of the word “look”:
Websters: “to ascertain by the use of one’s eyes”
Oxford: “to turn your eyes in a particular direction”
Dictionary.com: “to turn one’s eyes toward something or in some direction in order to see”
Seeing a pattern? They all seem to agree that looking is something you do with your eyes. Now, what does the Arcane Eye spell description say?
You create an invisible magical sensor that sends you visual information.
So the sensor doesn’t create an image for you to look at with your eyes (like a crystal ball)–if it did, then your friends could all look too. No, it sends visual information straight to your brain, the way your eyes send visual information to your brain. So with an Arcane Eye spell, your eyes, those things you must look with to be the target of a gaze attack, aren’t a part of the information chain.
In fact, the one official depiction of the Arcane Eye spell (formerly called Wizard’s Eye) that I could find shows the spell caster covering his eyes so that he can focus on the visual information being fed to him (it is from the D&D Expert Rulebook, 1E BECMI).
Granted, D&D 1E-BECMI is a long way from Pathfinder 1E, but the former’s Wizard’s Eye spell is a clear antecedent of the latter’s Arcane Eye.
So my interpretation would be that seeing something through an Arcane Eye is not looking at that thing and therefore is not a means to transmit a gaze attack regardless of the distances.