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Ok ok ok people...I just finished reading "Halo Grasslands" which explains how Halsey escapes Onyx. And seeing that I read "Ghosts of Onyx" as well I can safely say that Jun did die escorting Halsey.
If you purchased the limited edition or legendary edition of Reach the journal explains that Jun was killed by an elite, which may hint to it being the Arbiter, although I disagree with this theory due to the fact that the Arbiter and his fleet was chasing the Pillar of Autumn.
In Halo Grasslands, Halsey is trapped with Mendez, Tom, Kelly, Fred, and some other Spartan III's who were in the Onyx program being trained by Spartan-II Kurt (who died by self-detonating a nuke-head to save the others).
I understand that "Ghosts of Onyx" did not mention Jun because it was written during the days of Halo 2, but since "Halo Grasslands" was written after Reach's release (and serving as an introduction for the setting of Halo 4), Jun is definitely dead. He is in no way mentioned in Grasslands with Halsey.
TLDR:
Jun died escorting Halsey out of Reach, read the journal