The person (who is probably banned from this site for making such stupid accusations as this one) is incorrect.
Is it stupid? I expect anyone who hasn't touched probability in over a decade to give an incorrect answer like this one. And given the small sample of hands, it's not even very far off from the correct answer.
As the Wiz and others have pointed out, Alan. You have a 1 in 420 probability to hit quads and a 419 in 420 probability to not hit quads in any given hand. And the answer is 1 - (419/420)^10 = 2.36%. 2.36% is 1 in 42.55.
The reason why we calculate it this way with respect to the exponential, (419/420)^10, is that these are independent events always with the same result. Failing to hit quads. So (419/420)^10 = 0.9764 = 97.64% is the probability you will fail to hit any quads in the next 10 hands. The probability to hit at least one quad is simply all the other possibilities and the sum of all probabilities are 1. So hitting at least one quad is 1 - 0.9764 = 0.0236 = 2.36%.
The reason why it's slightly less than 1 in 42 to see any quads is that there is also a small probability you can get 2 or more quads in 10 hands. Hopefully you're playing "Shockwave Poker" when that event happens. This effect is more clear when you do the math near the 420 hand mark. The poster on your forum would say it's 100% likely that you will get quads within 420 hands, which is definitely not the case.
The probability of getting quads within 420 hands is actually instead:
1 - (419/420)^420 = 0.6326 = 63.26%
This is roughly the probability of getting a Royal within one "Royal Cycle" as well (approx. 40000 hands)
1 - (39999/40000)^40000 = 0.6321 = 63.21%.
Hope this makes things a little more clear for you and others.