In engineering Hiring good managers is not what they optimize for. Yes Engineering management should still have a high level of understanding of engineering principals within that domain, but management also because a people skill. Out of all my interviews, people skills and EQ are very often overlooked.
EQ has already been co-opted as a term and rendered essentially meaningless. One job, there was a craze for it, starting at the top, so naturally we all eventually had to go to this wretched little one day course on it. I kid you not, everything was about either "fear" or "love." If that sounds familiar, if it sounds like Donnie Darko , you would be right ... almost everything we were "taught" in this emotional intel…
EQ would be more about using empathy to connect to others, try to understand other's points etc. Of course, if you "teach" such stuff to manipulative people, you get wretched results.