I'd still say a positive claim should be expected to give more proof than a negative one. I mean, to prove "You have no house", I would have to check through an entire list of residences in the world and show that none were registered under your name, whereas to prove that you did have a house, all you would have to do would be show me it, and that you have the deeds to that house.
Of course, it would be a lot harder for everyone if you said your house was invisible, intangible, did not have any documentation and interacts with the world at only arbitrary points, then it would be very difficult for anyone to prove anything.
That is to say, it should really be expected for somebody with a positive claim to have absolute proof of it, and especially with existance claims, a negative proof would entail a sweep of the entire universe, which may or may not be infinite. Other things such as colour would be simpler for both parties though, where the biggest doubt would be vision problems with one of the participants.
It still to me just seems to make more sense to have the person making the positive claim provide evidence. Because, there will only be one circumstance under which something will be perfectly true, but many ways that same thing could be wrong. To prove something positive, it only needs to show that one example. To prove something negative, every single case needs to be shown as not being the case.