Not sure about your question.
As I understand the question there is an audio cable coming from the Chromebox that splits into two audio ports (3.5 mm) so two different people can plug in their respective headsets.
Is that correct?
Is the mute button common to both users (meaning that when the button is pushed both user headsets are muted)?
If the muting happens at random then the mute button is likely defective.
However there could be other reasons especially if the people are using their own and differing headsets.
That gets into TS, TRS, and TRRS.
FYI:
https://www.cablechick.com.au/blog/understanding-trrs-and-audio-jacks/
What that all means is if audio connections are mixed and matched some combinations, via the Y splitter, could be problematic.
More information needed.