The position of atheism is that a person does not believe in any gods. An atheist does not accept the claims being made by believers in gods. This is not a positive claim at all, but rather a rejection of claims being made by believers.
I'm not mincing words when I say I don't believe as the other person suggested. It would be silly to say that no gods exist, because it's impossible to disprove something and there's no way I can possibly know this. It is certainly possible that a god exists, but I don't see any evidence suggesting that any do exist. So, I feel that a very reasonable position to take is disbelief and rejection of the claims being made by believers. This is no mincing of words.
The burden of proof, as always, is on the person/side making the positive claim (i.e. God exists). The person who doesn't accept the claim need not demonstrate that the claim is false (again, it's impossible to disprove something). If I claimed that I can fly, you wouldn't accept me saying "Prove that I can't fly."
Atheism is a response to claims being made by believers, not positive claims being made by non-believers. This is no sophistry.
For more on the burden of proof, please read a previous post of mine here.