Do civilians panic?
Here's a few reasons to prioritize hunting aliens over saving civilians.
Losing a soldier hurts your grade about as much as losing 3-4 civilians. You should never risk 1 soldier to save 1 civilian.
Civilians hardly move at all. If they are behind your frontline, they are not in danger.
If civilians are ahead of your frontline, they can soak enemy actions that might be harmful to your soldiers.
While you're saving civilians that aren't in danger on this side of the map, the aliens are killing the civilians on the other side of the map. Losing a civilian to rescue a civilian that you were going to get anyway is not a good trade.
Here's some cases where you may want to save the civilian anyway
Explosives (yours or the aliens').
Very high health soldier with civilian near chrysalid that you don't have enough actions to kill. Deny that chrysalid a kill.
This way, there's no zombie.
- Last civilian standing. Apparently, bad things happen if all civilians die.
PS: Chrysalids don't care about cover. They don't seek it, and their melee attacks ignore it. Keep your squad close enough to support each other, even if they have to leave cover to do it.
No, during terror missions civilians will occasionally move in a feeble attempt to get away from enemies, but if left alone they don't tend to care. They're not armed, and will not take any offensive action towards your soldiers or the aliens.
I've neglected to save a civilian within a single move's distance from the Skyranger before, and he just sat there, cowering in fear, for the entire mission. If they're isolated, and no patrolling alien groups encounter them, they will be fine.
On the flip side, if left alone near Chryssalids, civilians can be turned to zombies that will eventually become Chryssalids if not killed quickly. This can mean your job gets quite a bit harder if you neglect the civilians too much.
Civilians in a way do panic but not in the same way your men do. On occasion an enemy can end their turn rather close to a civilian or someone can die near by that can cause a civilian to run away. How the AI decides where to run and I'm unsure but in a sense they do panic and run from their current position.
Despite this it is better to take your time and keep your men alive over having them die to save the civilians. Saving more civilians increases the panic reduction of a successful mission. It's based on the grade at the end of the mission (Poor, Good, Great, Excellent) of how many civilians lived.
As long as you don't fail the mission, it's worthwhile to still take it carefully and ensure all your men get out alive. The extra panic reduction on only that one country isn't worth losing seasoned men.