For those of you experiencing this too; after resetting the phone a few times (those of you who aren't sure what I mean when I say "reset" I only mean holding the home and power buttons until the screen blacks out and then the apple symbol appears, then letting go of those buttons-I am not referring to a restore) and then finally getting to the point of annoyance enough to do a full restore--I ended up having to call Applecare today. Resetting the phone got rid of the gray box every time...for about 24hrs. When I did a restore, it got rid of the box for about 4 days. Until last night....
Today I called Applecare and was advised to do a full restore again BUT set it to a "New Phone" not "install a backup" from iTunes/iCloud etc. So off I went to plugging in the phone to iTunes, hitting the "Restore this phone" and then proceeded to setup "new phone." I then signed in to my iCloud and it installed my contacts and calendar. Then I went to iTunes and added my apps and music. Clearly the phone was not set up the way I had it so it's taking me some time to go through it but it's just part of the process & one I won't currently be disappointed with, unless it doesn't work. Right now the gray box is gone (again) and I have hopes it will be gone for good. I'll keep you posted.