Yeah, except Halsey met up with the remains of Red Team and escaped. She also blew up CASTLE to keep the Covenant from getting any sensitive data. So, no.
Reach doesn't follow the books at all though (which is dissapointing in my opinion, not respecting the fantastic work Nylund has put into Fall of Reach, First Strike and Ghosts of Onyx and replacing it with a less impressive story) meaning that anything could have happened after.
Also, wiki pages aren't reliable at all. They're just edited by people who are speculating as much as us. Just because its on Halo Wiki doesn't mean its true. All we know for certain is that we didn't see Jun die.
Actually, Bungie does cover their asses pretty well here. Bungie On The Contradictions Between ‘Halo: Reach’ And ‘Halo: The Fall Of Reach’ Novel » MTV Multiplayer
Jun's annoying talking caused Hasley to go berserk a crash into something that has to do with chocolate. I really dont know what happened, my theory is that one of them was hurt and crashed the pelican, similar to what happend to carter
I think were all REACHing here, pun definitely intended. I think we need to review a good point, the whole time you play the campaign from the first playable mission is that your on a CONTINGENCY mission. This means doing whatever it takes to stop something, ie the covies from getting anything they can use against the humans, So A SPARTAN sacrifices himself and kills Halsey to do so. Its heroic and cliche. Just think of what would happen if this were all real. EPIC!
Regardless, it's been quoted MANY times in Halo 1, 2, and 3. They ALL say that Master Chief is the last spartan alive. Well, maybe not ALL of the Halo's...
Jun isn't awesome enough to have survived though. Although on a second though he does seem like the kind of guy who would hide out for years to save his own ass.