So why are people talking about vulnerabilities and 'timing attacks' then? and how are you supposed to get the square root of a number you dont know..

You don't need to use timing attacks or other vulnerabilities. Others have pointed out how to do it, use SPI and provide your own implementation. The xkcd post is probably the best hint ;)

The private function divides the unknown square by another number you provide. It's possible that how long this division takes depends on just what the secret number is -- keeping track of this and carefully feeding it input might reveal the number. That's a timing attack. (This is something you really have to consider when designing cryptographic functions and other such hardcore stuff.)

I suspect that doing something with the random seed is more what the author had in mind, though.

From what I understand a timing attack on SecureRandom to find out which number it generated. But I might be wrong in which case I'm sure someone knowledgeable will give you a better answer:)

edit: Yeah… just ignore this and look at the other answers ;)