First the whole burden of proof thing:
This only applies in a court of law. In every other situation both side in an argument have to argue their positions and the one with the stronger arguments wins. Saying the burden of proof is on you is just childish in a situation like this.

"I can't prove a negative."
That's one of the most stupid sentences EVER!
Why not? I is not so hard to "proof" that the sun doesn't revolve around the earth. Another negative that is even easier to proof: I didn't kill JFK. I wasn't born when he died.
I can only redeem the sentence when I delete the "a negative" or I add "nor a positive".
In scientific (and philosophical) terms you can't proof anything. You can only disproof a theory or statement.
An experiment can work a million times and it would still proof nothing, because it is possible that it doesn't work the next time.
When people (in Britain) wanted to say something is impossible they used to say:"That's as likely as a black swan."
They have all seen swans. They were always white, and they "knew" that the next swan they were going to see would be white again. This observation has worked billions of times.
They were extremly sure that swans have to be white.
Then Australia was dicovered and a little later black swans.

"The first thing Fascists usually try to do is silencing the opposition."