You cannot move the eye into the antimagic field.
The eye is not a creature, and spells typically tell us when they create objects, so we conclude that the eye is just a magical effect. Now, we see in the antimagic field description:
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.
The eye, being a magical effect, cannot protrude into the antimagic field. Based on this, I would rule that you cannot even move the eye into the field, it should be blocked from entering the field at all. Obviously, this isn’t definitively explained in the rules here, but I think this is the most reasonable ruling based on what we’ve got. Your mileage may vary from DM to DM.
If the field is created around the eye, you cannot move the eye out.
Moving your arcane eye is a function of the spell:
As an action, you can move the eye up to 30 feet in any direction.
Since moving the eye is an effect of the spell, you cannot move it if the field is created with the eye inside it:
While an effect is suppressed, it doesn’t function
Thus, the spell effect of moving the eye does not function while the eye is in the field.