Flight Instructor with student reported an NMAC in the traffic pattern. While the instructor was descending over the airport another aircraft not communicating on CTAF flew directly beneath them.

Narrative

We were flying over CKN at 4500 ft. to set up for a simulated engine failure at altitude and make a power off landing on the airport. We were making a good amount of radio calls to let the airport CTAF know our intentions and scanned the area for traffic before we started descending. As we started a circling descent over the airport a yellow aircraft was holding short of Runway 13 and asked if they had enough time to takeoff before we came in. We let them know they would be able to takeoff because we still had a good amount of altitude to lose. As we were watching the yellow aircraft takeoff we were about 3000 ft. MSL and made another radio call while continuing the circling descent. All of a sudden a high wing aircraft (white with blue stripes) flew directly beneath us from the south; heading north; and made no attempt at making any radio calls. That traffic was not on ADSB either. I took the flight controls and started a climb back up and let the airport CTAF know that we were stopping the simulated engine failure. We watched that same white high wing aircraft with blue stripes continue flying north and seemed to have no intention on actually landing at Crookston. We decided to fly back to ZZZ after that event.[Caused by] an aircraft flying right over the Crookston airport at lower altitudes and not making any radio calls or listening to the airport CTAF.[I suggest] more information to general aviation pilots about making radio calls and listening to the CTAF frequency when near an airport to gain situational awareness about what is going on.