Flight Instructor reported taking evasive action to avoid an aircraft flying a non standard traffic pattern resulted in an NMAC.

Narrative

I am a CFI working with new student pilot on takeoff; pattern and landing with a slight crosswind condition on clear VFR afternoon. We started takeoff roll; Runway 2; then Aircraft Y announced 10 miles to the north. We proceeded upwind to 500-600' AGL then announced crosswind for Runway 2 and continued climb. Aircraft Y announced 5 miles to North we were had pushed button to announce downwind and saw Aircraft Y bearing down on us at same altitude and direct at us. We were only at Vy but pulled up and Aircraft Y slid under us maybe 20' directly below us...we were both shaken but continued downwind at 1400' AGL. The Aircraft Y still had not made a downwind announcement. We asked if he saw us and he said no but looked like he was downwind; base and final at 500-800' AGL.We landed the airplane then I spoke with Name; about the incident. He said he was just returning from ZZZ under the mode C veil and we were not where he expected us or where ADS-B was showing us! I was in disbelief - when I quizzed him about patterns he thought crosswind was only the width of the runway and did not understand why we were in his downwind path. I tried to explain it to him but he told me it was our fault for being in his way and turned and walked off. I encouraged him to quit looking at all his glass and get a view outside his plane.