AFAIK squeeze is usually associated with pressure changes, so once you're at a certain depth the pressures in the sinuses should have equalised with the ambient pressure. Does this happen on ascent/descent? If so, it should subside if you go in the opposite direction.
On the other hand, do you get
Ice-cream headaches when eating ice-cream or drinking very cold drinks? I do and the headache is very very similar to the one you describe - a sharp pain occurring in the middle of the forehead. For me it goes away in a few tens of seconds, so again it's similar to your experience.
The air you breathe underwater is quite cold, a few degrees above freezing. I can imagine that breathing it could cool down the back of your throat enough to trigger the headache. Especially since you mention that it occurs more frequently in cold water. The lower the water temperature, the lower the temperature of the inspired air. Does the headache occur more frequently if you're swimming harder - thus breathing more air?
Try drinking a large amount of iced water and see if you can reproduce the headache on land. If you can, you could try using your tongue as a guard, as PADI recommends when you take a regulator from a buddy to guard against inhaling water, and heat up the air. But then your tongue might freeze and fall off
