From a philosophical point of view, the burden of proof is always on the person contending that something does exist. It's the basis for empiricism/the Scientific Method. Beyond that, it's essentially impossible to prove a negative and if we required that, we would have to believe that anything we can't prove exists, does exist. That would seem to me to have all sorts of odd consequences. For instance, I could claim there was a teapot at the center of the universe that created the universe. You can't prove their isn't therefore, if we require proof that something does not exist, you would have to believe that that teapot does exist and it created the universe. The same would apply to any nonsensical thing anyone could think of.