If your soldier gets gravely wounded in a mission, there is a chance that the soldier gets shaken. This chance is 20/25/25/30% for the four difficulties.
What is gravely wounded? It means that the recovery time is more than 168 Hours (336 on Legendary) - this is determined at the end of the mission.
How is recovery time determined? Based on the lowest HP percentage at any point in the mission, a number is generated within certain limits. There are four "buckets": up to 20%, 21-50%, 51-75%, 76+%.
Example, your Soldier has 10 HP and takes 6 Damage, which brings him down to 40% health. At the end of the mission, the game looks in the 21-50% bucket to see the range of how long recovery takes and picks a random number from that. If that random number is more than 168 hours (336 on legendary), you're gravely wounded and the game rolls if you're shaken.
Note that the source of damage doesn't play into it, and neither does healing. The game records the lowest your HP ever was. E.g., 10 health, take 6 damage, heal 4 damage, take 3 more damage, heal 4 damage -> Even though you end the mission with 9 HP, the game records that you were down to 4 HP, which is 40% of 10. That percentage is the ONLY thing that counts.
I do note that the spread between best and worst recover time is MASSIVE. For example, a single point of damage (the 76%+ bucket) on Rookie difficulty puts your Soldier out of commission for 5 to 25 days - completely random.
For more technical details, I dug into the source code here.
Bonus Answer: If you recover from being shaken, you get a permanent increase to Will (4 to 13 points) and can't be shaken again - it's like Vaccination.
I have seen these before and opened them a number of times.
On each occasion the box is visually empty - as in your screenshot. But the audio of the solider says "there is some loot here" and there is a sound of items been scooped up.
I just did a VIP extraction mission on L/I - and I noticed I had an "extra" item at the end. I had 2x repeaters, an elerium core, and a scope. That is 4 items, but I only ever get a max of 3 from an alien drop.
So I can assume at this point the 4th item came from the chest. This would imply you get a "random" item from the chest at this stage...
Edit: I can now confirm - you do receive "random" items from the container. I just completed a mission, and received an extra "Laser Sight" and "Hair Trigger" mods that were extra to what I had picked up in the mission.
Best Answer
There is a templar resistance order called Noble Cause that can boost the will recovery speed of all soldiers by 20%.
Unfortunately, you can only assign this at the end of each month so it may not be possible in your situation.