Just had this problem when putting screen back on an iPhone 6. 3 Sets of parallel white boxes on the top half of the screen. Miki's solution was very helpful -- thank you. The white boxes went away after about 8 hours after a full charge. Two things I would add: 1) test if the touch screen works completely, including in the areas with the white boxes and 2) test if with the battery removed (not just off), whether you can see the white boxes faintly in the display. If these are both true, it is likely to be static (or some charge) in the digitizer rather than a damaged or incorrectly seated connection that hopefully works itself out with time. The most perplexing part for me was the white boxes when the screen was off (and removed) meaning that there was something with the digitizer rather than the cables.
Thanks again for the comments on this page -- really helped!