When the second Hunter showed up, why didn't Halsey run inside CASTLE base when she was by the door? Because she saw Jun coming from behind, and so chose to distract before it turned around and found out.
Jun's injuries from the Hunter have got to be pretty severe, considering how far it hit him. Throw that in to him refusing any medical care, and more likely than not, even if the Covenant don't get to him he may die anyway as a result of internal bleeding.
So how did Jun deduce Eliza Ingrid was the real rebel?
He's been watching her closer than anyone else outside of the Innies, and the same issues that made him border on obsession with a target (which he overcompensated for with an "unhealthy amount of detachment") also gave him better psychological insight into her.