It could be that they have some repeat customers who need to pay with a company card, and they want to accommodate those people but not the general public. The expense account customers may place larger orders for meetings etc.
I believe that would be the case with this restaurant, but I don't think the fast-food place has anything like this going on. It is in the food court of my local university, so I can't imagine they have many customers wanting to pay with AMEX anyways.
Next time I am going to try paying with it by inserting it instead of my Visa, as the terminal doesn't require the card type to be specified (I don't think).
Is it worth reporting the business to AMEX? Or will they likely do nothing?