lol, I feel like I'm about to sound really picky here, but I'm a logic instructor at a university and I can't keep my mouth shut on this one :evil_lol:
This is a common claim about proving negatives that doesn't quite capture the scope of the problem and so can be misleading.
I'll just say that in standard deductive logic it is perfectly possible to prove a negative statement.
As for the issue of burden of proof for existence, some negative statements are also quite provable. For example, "There are no married bachelors" is provably true, due to the contradiction in terms.
There's a pretty good wikipedia article on the issue, in logic and in other fields, if you want more info:
http://skepticwiki.org/index.php/"You_Can't_Prove_a_Negative"