The burden of proof always rests with the one making the claim, and especially if it's an extraordinary claim.
However, JWs are so indoctrinated that they unashamedly throw logic, reason and critical thinking under the bus, calling it "worldly wisdom", "independent thinking", "satanic propaganda", etc, in their bid to defend their organization and its foolish teachings and policies at all cost. Sam Harris once made this statement (and I'm paraphrasing):
"If someone does not value reason and logic, what logical argument can you use to demonstrate to them that they're wrong?"
I think best way to reach a JW would have to be to first get them to see the value of logic and critical thinking - without bringing up their religion, or even the subject of religion at all. You have to get them interested in the subject of critical thinking and its value in helping one sift truth from fiction. Maybe you can do it in the context of avoiding scams by unscrupulous "worldly" con men and the spin of "worldly" politicians. Then maybe, you can talk about the psychology of those who fall for such scams - how they are denial and make all manner of ridiculous excuses even when shown evidence exposing the scam. You basically have to get them to learn how to spot BS and how to spot the BS excuses by those taken in. Then you can gradually shift over to looking at the members of false religion, highlighting all the kinds of fallacies and excuses that religious people give to defend their continued loyalty to false teachings and false religious organization.
So first, you get the JW to value logic and critical thinking. Next you train him to spot the fallacious coping mechanisms the deceived people use when confronted with the possibility that they've been deceived. Then you relate it to the subject of false religion. Finally, if you think the JW is ready for it, you relate it to his own religion. But this final step is optional. If you do a good enough job with the previous ones, you can probably trust that the JW will figure out this last step on his own.
So JWs have the burden of proof. But we have the burden of logically proving that fact to a group of people who value emotionalism and blind loyalty over logic.