When I export my animation with a white background it shows that it has a black background in Premier Elements.
I exported in in a SWF file.
I am not sure how the relationship between Flash and Premiere works, but if it is anything like putting a Flash file in another Flash file then the loaded Flash file will not have a background. What you should try doing is to draw a white rectangle on the bottom-most layer of the Flash file timeline that covers the whole stage.
It's because it's seeing the alpha channel, so essentially there's nothing there which is why it's black. Put a background on the bottom most layer and it will be fine.