Flight Instructor reported a NMAC while in the pattern at a non towered airport. The instructor; with trainee; took evasive action to avoid the small business jet.

Narrative

I was flying with my student in the pattern at ZZZ working towards the completion of our flight syllabus about learning to land and utilizing our flight school aircraft. Winds were favoring Runway XX and thus we were making left pattern traffic for Runway XX. Upon takeoff for our 4th trip in the pattern; we identified a company aircraft on the localizer approach for Runway YY. This traffic stated that they would break off the approach and enter the left downwind for Runway XX. I instructed my student to depart; referencing the position of the traffic on my electronic flight bag (EFB) and determining that we could cycle into the pattern safely behind them. After rotation; we heard another aircraft call on approach for Runway YY. I made a radio call stating that we'd just departed Runway X and would be turning left crosswind for Runway XX. At this time the TCAS was informing me of the traffic directly ahead of us approaching Runway YY (head on with our current heading). The aircraft on approach then made a call stating that they would enter the downwind for Runway X 'if that's what you guys are doing.' This aircraft identified as Aircraft Y and was piloted by Person A. At this point; I took the controls from my student and made another radio call stating that we were in the left crosswind for Runway XX. Aircraft Y broke off from the Runway YY approach course and flew directly at us at pattern altitude. I reduced power and dove to 1600 ft. MSL to avoid a collision with the Aircraft Y. The aircraft flew directly over us; TCAS indicating that they were within 300 ft. I leveled the aircraft then climbed back to pattern altitude (1;800 ft. MSL) and exited the pattern to the west with the intent of coming back to enter the left downwind to Runway XXI asked Aircraft Y if they had seen us visually or on 'ADSB'. He answered in the negative and asked if I had seen him. I expressed that I'd watched him fly 'right over us'. He offered to extend his downwind to allow us to land; to which I told him I was 'leaving the pattern to the west to teach my student how to re-enter the pattern.' I then asked him how long he'd be at the airport with the intent of speaking with him about the incident. He said he'd be at the FBO for more than an hour. We then rejoined the pattern and landed on Runway XX without incident. After landing; we secured the aircraft. I then went and spoke with Person A; the pilot of Aircraft Y. The exchange was friendly and professional. I explained where I was in the pattern when the incident occurred and why I perceived it as a dangerous situation. He apologized and stated that he never saw us. He also stated that the only reason he decided to use Runway XX was because Approach had told him that Runway XX was the runway in use. He believed that Runway YY was the preferred runway for when winds were calm (METAR for the time of the intendent shows winds 020 @ 5 kts). I corrected Person A; explaining that while Runway YY has instrument approaches; Runway XX is the recommended runway during calm winds. He also made the note that the winds seemed to be about 060. Our conversation ended with a hand shake and a pleasant farewell.