It is unknown. He is a Spartan that is truly MIA. He is last seen escorting Dr. Halsey to CASTLE Base.
most likely though because he is a dumbass camper who sits there and yoinks you all day
Jun does not die in the game Halo:Reach, as far as we know. He is last seen in-game escorting Dr. Halsey to CASTLE Base via Pelican. Its very possible and perhaps likely though that he did die at some point after we last see him on Reach, for Dr. Halsey and Jun are not together when Halsey is rescued by Blue Team's Spartans in the novel First Strike. It is also possible, although unlikely, that Jun escaped Reach via some other means. I'm sure it will be revealed to us in a future novel or the like.