It's very odd, I never received this message in the past, but I recently booked a WT+ to CW/CE MFU return itinerary from IAD->LHR->CPH for Ms1995hoo and me (we're going on a cruise) and entered all of our info and got the same error message. The only blank space left to be filled in is the Green Card number, which of course we do not have as US citizens. It appears to me that BA's system cannot understand that it should grey out the Green Card box for someone who is a US citizen and that this is probably why those of us on this forum who are from the US are having this issue--the system sees an empty field and flags it.

Other carriers' systems seem to have bugs too, though. I'm flying on VS in Upper Class next month and their system made me enter an address, phone number, and reason for travel to the US, even though I told it I'm a US citizen (and thus am not required to enter this nonsense). So it seems the problem isn't totally unique to BA.