By being born in the 90's I do not know how serious of a concern was for the average persons, but on the other hand, I can give some information about lesser known conflicts or some possible conflicts in the region. We must keep in mind that Eastern Europe and the Balkans(especially the Balkans) are a mix of different nationalities and while the assimilation policies, forced assimilation or plain population exchange that happened since the XIX century did not make the region homogenous in this regard. Even now, there are a lot of minorities(ethnic or religious) in these countries.

That being sad, I it easy to understand why a spill over was possible, especially since in the same time we have another geopolitical change in the area: the breaking up of the USSR. We must also not forget that some communist regimes developed nationalistic tendencies in the 80's(e.g. Romania, Bulgaria, Albania).

Now, for some other conflicts. In the early 1990, so just after a few months since the Romanian December revolution, there were ethnic tensions in a large city called Târgu Mureș between Romanians and Hungarians(the city itself is half Romanian, half Hungarian). That could, in worst case scenario, extend to a civil war between Romanian and Hungarian population in Romania with an intervention from Hungary.

Another conflict in the area is the Transnistrian one. There was open battle which could spill over in let's say the rest of Moldova.

Regarding Bulgaria and Turkey. I do not know that local conflicts that could spill over, but during the 80's Bulgaria there was a Bulgarisation campaign against the rather large Turkish minority and Bulgarian nationalism was rather high. Harari was concerned here that after the fall of communism that nationalist rhetoric and practice(now without the restrictions of the former regime) will cause a reaction from the Turkish minority and ignite an internal conflict. In such a scenario, Turkey will have had intervened to "protect" the Turks in Bulgaria.