A Medusa with a Hat of Disguise

If a Medusa had a Hat of Disguise and used it to look like a Morlock would her gaze attack still work?

Or would the disguise self spell that the Hat of Disguise count as a veil?

The creature can also veil its eyes, thus negating its gaze ability.

Could the medusa choose to have it count or not count?

That's a really good question (and one I hope they touch on in the next Classics)!

RAW, I believe it would still work. Disguise self is an illusion and doesn't shut down racial abilities like alter self does. (example a drow trying to look like an elf still has darkvision, not lowlight vision.

Now that being said... it really depends on how the medusa's gaze attack works. If she's like the original, then it's that she's so damn ugly and the hat would work (though really she shouldn't be able to shut it off on her own. How do you control when you're ugly enough to petrify or not?)

If it works like a gaze attack ("Look into my eyes, little gnome, and end up on my lawn!"), then shutting it off makes sense. Eberron's medusae work like that. I believe it's fluffed as closing the humanoid eyes and seeing through the snake eyes.

The medusa has to cover its eyes for it to not work. Changing her appearance does not shut off supernatural abilities. The had of disguise would not change her into a moorlock. It is just a disguise. Her eyes are still functional.

If a Medusa had a Hat of Disguise and used it to look like a Morlock would her gaze attack still work?

Or would the disguise self spell that the Hat of Disguise count as a veil?

PRD-Gaze Attack wrote:

The creature can also veil its eyes, thus negating its gaze ability.

Could the medusa choose to have it count or not count?

Mechanically it is a gaze attack which takes a standard action to activate and focus on a target. It has a choice on when it is brought into play. If disguised via the hat, it still keeps that option. The ability isn't an aura or an ability like the nymph where the opposing creature looks at them and it takes effect. Veiling its eyes could just mean it squints for all it matters mechanically, it doesn't mean it has to wear an actual veil. The wonder of fluff text can make it sound a number of ways without it it affecting the mechanical aspect.

Petrifying Gaze (Su)

Turn to stone permanently, 30 feet, Fortitude DC 16 negates. The save DC is Charisma-based.

Versus

Blinding Beauty (Su)

This ability affects all humanoids within 30 feet of a nymph. Those who look directly at a nymph must succeed on a DC 21 Fortitude save or be blinded permanently. A nymph can suppress or resume this ability as a free action. The save DC is Charisma-based.

Gaze (Su) A gaze special attack takes effect when foes look at the attacking creature's eyes. The attack can have any sort of effect: petrification, death, and charm are common. The typical range is 30 feet, but check the creature's entry for details. The type of saving throw for a gaze attack varies, but it is usually a Will or Fortitude save (DC 10 + 1/2 gazing creature's racial HD + gazing creature's Cha modifier; the exact DC is given in the creature's text). A successful saving throw negates the effect. A monster's gaze attack is described in abbreviated form in its description. Each opponent within range of a gaze attack must attempt a saving throw each round at the beginning of his or her turn in the initiative order. Only looking directly at a creature with a gaze attack leaves an opponent vulnerable. Opponents can avoid the need to make the saving throw by not looking at the creature, in one of two ways.

Averting Eyes: The opponent avoids looking at the creature's face, instead looking at its body, watching its shadow, tracking it in a reflective surface, etc. Each round, the opponent has a 50% chance to avoid having to make a saving throw against the gaze attack. The creature with the gaze attack, however, gains concealment against that opponent.

Wearing a Blindfold: The foe cannot see the creature at all (also possible to achieve by turning one's back on the creature or shutting one's eyes). The creature with the gaze attack gains total concealment against the opponent.

A creature with a gaze attack can actively gaze as an attack action by choosing a target within range. That opponent must attempt a saving throw but can try to avoid this as described above. Thus, it is possible for an opponent to save against a creature's gaze twice during the same round, once before the opponent's action and once during the creature's turn.

Gaze attacks can affect ethereal opponents. A creature is immune to gaze attacks of others of its kind unless otherwise noted. Allies of a creature with a gaze attack might be affected. All the creature's allies are considered to be averting their eyes from the creature with the gaze attack, and have a 50% chance to not need to make a saving throw against the gaze attack each round. The creature can also veil its eyes, thus negating its gaze ability.

That first bolded sentence means when you first enter gaze range you make a save. The monster can also choose to force another save as a standard action. This is covered in the next to last paragraph. I bolded the relevant sentence. On your turn you also have to make another save if you are in range. This is covered with my second bolded sentence.