An Antimagic Field is a 10-foot-radius invisible sphere which suppresses the magical effects on objects and creatures, prevents spell casting inside its AoE and spells with an AoE (e.g., Fireball) can't extend into the sphere. Moreover, magic items become mundane. In particular (emphasis mine)
Spells and other magical effects, except those created by an artifact or a deity, are suppressed in the sphere and can't protrude into it.
Concerning summoned creatures and objects, the description say:
Creatures and Objects. A creature or object summoned or created by magic temporarily winks out of existence in the sphere. Such a creature instantly reappears once the space the creature occupied is no longer within the sphere.
The description of Arcane Eye says:
You create an invisible, magical eye within range that hovers in the air for the duration.
Nowhere it is specified that this invisible eye is a creature or an object, but since its description says that it is magical I believe that it is suppressed by the Antimagic Field given the first quote about this spell (see the SAC for determining if something is magical).
But once the eye entered in the AoE of the Antimagic Field can the caster move it out the sphere via the mental control? Or must one apply the same rule for objects and creatures?