Max,
Post a picture or video of you setting up the squat "SS-style." Let's see where your wrists are.
I would strongly recommend you try to keep the double overhand grip for as long as possible over as many sets as possible. If you have no issues holding the bar then keep the double overhand grip. I didn't have to switch to alternating grip until around 345 lbs (and I'm no genetically gifted specimen by any stretch of the imagination). And it makes me angry that it was that early.
Curious-- does the pain you experience from this squat grip problem happen to be in the same arm that's supinated during the deadlift?