ITS 12 rules for HVTs lists them normally as "Neutral Civilians" (p.14) so it seems that normally either player could Synchronize with theirs or the other player's HVT and enter a Civevac state to move them around (I'm actually unclear on this because the Synchronize Civilian and Civevac text mention just "Civilian" in what seems like it could refer exclusively to the "Civilian" figure/statline itself and not to the quality/state that HVT has of being a Neutral Civilian) The rules go on to say that if an HVT is a Designated Target, "they are considered Enemy Troopers instead of Civilians" and that they are "reactive and hostile, reacting as if they are an Enemy Trooper" (p.10) So if an HVT is a Designated Target they are no longer categorized as Neutral Civilians. However, that's pretty much the beginning and end of what the book describes, such that it's very vague on what other things change as a result of this. Namely, the rules say that a Designated Target is reactive, but does not say whether or not they can now be activated with orders (one assumes that they can't because they aren't part of a combat group and so who knows where the orders would come from). If they are no longer Civilians, though, can they still be synchronized with (by a friendly figure) to allow a player to Civevac and move them around to hide them from the enemy? If not, is there any method by which a player can move their Designated Target figures aside from an ARO dodge?
The enemy HVT is considered a hostile trooper in those scenarios. They'll still be neutral civilians for the "friendly" side. And no, they won't ever be troopers part of your army lists, so you can't spend orders on them. Just remember that a CivEvac civilian is limited to Short Movement Skills only; that precludes BS Attack, Dodge and Reset. If you synch them, they can't offer even token resistance when attacked!