Flight Instructor on training flight with student reported a NMAC with another aircraft in cruise.

Narrative

I was PIC conducting flight training in Aircraft X. I had to take controls from my student and sharply evade an aircraft; Aircraft Y; to avoid a head on collision. We were approximately 5 miles East of the regional airport headed Northeast bound towards the VOR and we had just leveled off at 3;500' MSL on our way to ZZZ1. We had departed the pattern and made all proper radio calls. After I evaded Aircraft Y; the pilot of that aircraft made his first radio call to the CTAF at ZZZ which my student and I were still monitoring. I informed the pilot over the radio that I had evaded him. His response was 'yeah; I saw you too.' He was traveling Southwest at 3;500' to land at ZZZ. His aircraft appears to have had no ADS-B out; as I was monitoring traffic on my EFB in addition to seeing and avoiding other aircraft. I never heard Aircraft Y make a 10-mile radio call on 122.7; the CTAF Frequency at ZZZ.It came to my attention that the PIC of Aircraft Y was in conversation at a FBO. CFI discovered upon review of aircraft; pilot logbooks and records; that the pilot was flying with an expired medical certificate and without proper student pilot logbook endorsements. Additionally; the CFI of the pilot had not logged proper flight training. CFI had not endorsed logbook for a pre-solo knowledge test despite endorsing him for solos; nor has he properly maintained said 90-day solo endorsements in the several years that he has apparently been verbally releasing the pilot for solo flights in Aircraft Y. This behavior of arriving to the airport and 'receiving verbal permission to fly...' was corroborated by multiple individuals of the line staff a FBO.