It's been a while since I jacked around with the math, but here goes.
1st Ace 4/52 = .0769
2nd Ace 3/51 = .0588
1st King 4/50 = .0800
2nd King 3/49 = .0612
1st Queen 4/48 = .0833
2nd Queen 3/47 = .0638
Multiply the odds of each one of theses events to derive the odds of the combined event = .000000118 = 1/8,482,716.6666
This would be the odds of calling this shot beforehand for any three players, but at a full table of 9, and this is where I may need some help, I believe the odds of this event occuring are at least 3 times greater or 1/2,827,572.2222. Still a big number. But how unusual is it? To put it in perspective it is about 39 times less likely than being witness to a Royal Flush. I play about 2000 hands per month including tournaments or about 24000 hand per year, lets just call it 28,275. I should see this event occur about once every 100 years.
ETA: The 3-handed probability you quoted is wrong because the person didn't take into account the 3! possibilities of which players get which pairs. So multiply by 6 and it's 1 in 1413786.
Though I would have done it like this: C(4,2)^3 / C(52,6) / (5*3)
