Obviously, the recruiters at the assessment day won't physically check your passport (not that it would do any good anyway, since your overstayed visa could have been in a different passport). They'll rely on your honesty. The website doesn't say anything? Go along on the day. If on the day you are asked to fill in some form of paperwork that asks about your passport status, then just be honest.
I severely doubt that it will be a problem for you, though. If you haven't been dishonest with them during the recruitment process, i don't think that they would fire you if when they request for you to get your US visa it is refused. Middle Eastern airlines' employee relations might be a little flawed in some respects, but even i would be surprised if i heard of that happening.
I don't think the passport issue is really that big of a deal for Middle Eastern airlines, since they recruit so many different nationalities.
Focus on actually getting the job offer, then sort out the details like this!
Again, good luck.