The mission is called
All Ghillied Up, and I think you are reading the situation wrongly. there are 2 helicopters in that mission (before 'the' sniper shot, which you won't have gotten to yet). Neither of them require taking down.
The first is when your following Captain McMillan and you approach a graveyard. He says "enemy helicopter, get down". He goes to the left hand wall, you should go to the right, and get down on the ground. Stay there until McMillan tells you that you have the all clear, then follow him again. The second helicopter flies overhead later. Again, ignore it, follow McMillans instructions and you'll be fine.
It is possible to take down the helicopter, but it is a waste of time. It doesn't achieve anything, and just takes up a weapons slot. And here's a hint - swap your pistol for an enemy rifle (preferably an AK47). Do not fire it (it isn't silenced), but you will need it on the second Ghillied up mission.
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