This part takes place during the second half of Rebirth Island while playing as Hudson, it will be pretty obvious as you and your squad are wearing the suits. This achievement begins when the gas drops and everyone puts on their helmets, up to the part where you are about to enter the facility; you'll make a drop where your squad and self are clear of the gas and you pull off the mask and the achievements should pop there.
Some tips are to mostly hang back some and use the thermal scope to pick off enemies from distance. Generally when you are hit, you lose some health and you regain it.. well this is not the case, your suit/mask will crack from gunfire and the more you are hit, the higher the risk the suit will break - almost like you are given a health bar with no regeneration. If you are still having issues, try it on an easier difficulty.
Now this part is very important: If you are having trouble and your suit does end up breaking, very quickly pause (before it does break and you fall down) > save and quit (since there is no reload last checkpoint). Should be able to resume from main menu at a checkpoint prior to when your suit broke and still obtain the achievement. If you wait for it to restart you at a checkpoint and then save and quit to restart it, it will not count since you need to get through this part alive in a single playthrough. The save and quit method is there to help register the single playthrough.
I want to make a note, this may not happen for everyone but I was pretty damaged at one part and when the time came to take care of the two helicopters my suit was "magically" repaired to a point. Was playing on Normal so not sure if this is intended or a glitch.