Quoted from the Instrument Procedures Handbook, page 4-6:
The approach clearance provides guidance to a position
from where you can execute the approach, and it also
clears you to fly that approach. If only one approach pro-
cedure exists, or if ATC authorizes you to execute the
approach procedure of your choice, the clearance may
be worded as simply as “… cleared for approach.” If
ATC wants to restrict you to a specific approach, the
controller names the approach in the clearance—for
example, “…cleared ILS Runway 35 Right approach.”
When the landing will be made on a runway that is not
aligned with the approach being flown, the controller
may issue a circling approach clearance, such as
“…cleared for VOR Runway 17 approach, circle to land
So if they name the approach, you are restricted to that approach.
I searched the rest of the book for items specifically related to non-towered operations, and found nothing that contradicted that.