IIRC, the instructions on how to use the EEO free night offer and get a reservation booked on that offer are listed on the back of the card. I'm pretty sure that if a reservations agent follows that, you can't get an AAA/AARP/GOV rate, etc.

Although you may get exceptions wherever you travel and get a rate not normally available when you use the EEO free night offer, that is just what it would be, an exception.