Yes, the black boxes will disappear if you leave the program running long enough. This program uses progressive iteration counts and progressive resolution which is why it does this. For points very close to the boundary it takes a long time to determine if a block is in the set or not, so it stays black and undivided for a long time. I did try making "not yet determined" blocks use the outside colouring instead of the inside, but it didn't work as well for various reasons. And subdividing the block too early (to try to get rid of the blockiness effect) would lead to a lot of wasted work if we end up subdividing blocks that are inside the set. Basically there are a lot of tradeoffs to making all this work - I chose tradeoffs which get you to the final image faster at the expense of making some of the intermediate images look "wrong".

I was also wondering why sometimes the rendering on this page leaves black boxes in the image, is it just not running long enough?