They eyes are an important body part in any strength exercise.
Why?
The eyes lead the movement of the head - and the head leads the movement of the rest of the body.
Have you ever seen a snake make its way on the ground? You can spot how its body follows the direction of its head.
Back to strength training: When you fixate your eyes onto a given point in the press, your head is being held still. When your head is being held still, you have better control over the rest of your body.
The reason for that is that there is less "interference" from one body part (i.e. the head) onto other body parts (i.e. legs, hips, shoulders and arms in the case of the press).
When you fixate your eyes forward in the press - like two beams of a laser pointer - your sense of balance and stability improves.
Eyes down in the press, and you don't know what's going on above your head. Eyes up in the press, and you loose spacial awareness of your hip, leg, and foot position.
There is a tendency - especially in the last one or two reps - to divert your gaze away from the focal point in front of you in the press. As things get hard, technique has the potential of getting shaky.
Override this temptation and make your gaze stick.
Details matter.
As do your eyes.