If Eugene’s recorder caught everything that Dwight had to say and he shares the recording with Negan, Dwight would almost certainly be killed for his betrayal.
Also, given Eugene’s history of indecision, he may be the character in danger if Dwight finds out about a recording.
Now, on the one hand, Eugene’s recorder didn’t exactly look state-of-the-art, and Dwight was speaking from a distance.
Dwight probably would not hold back from killing Eugene if it meant saving lives and taking down Negan, and the same might not be said about Eugene turning in Dwight.