The campaign itself is pretty straight forward on Normal and Hard so I'm going assume that those who are looking for a guide are the ones playing on Hardcore for the achievements. This guide should still apply to the lower difficulties anyways.
I do not remember much of the first mission of Under No Flag since I beat it when the game first came out, but from what I do remember, it's a pretty straight-forward mission. I don't recall any one thing being obscenely difficult. I would play through this mission and get a feel for how difficult things are going to be. You'll find out pretty quickly, especially on the higher difficulties, that the enemies are pretty accurate. Test out some different strategies and find out what's working best for you. If you die, no biggie. The first level is pretty short anyways.
The second level of Under No Flag is the hardest and the longest without a doubt. Why in God's name you spawn with the gun you spawn with I don't have a clue. It's a short range rifle and this is a wide open area. Sucky weapon choices aside, there are three different objectives. I did, and would recommend doing them, in this order.
The first point you'll want to hit is the communication building. I found it to be the least guarded. There is still a decent enemy presence though. Above the building is a mountain with a sniper's nest. If you go around the mountain and come up from the other side of the comms building, you can assassinate the sniper and take his silenced gun. Now, silenced doesn't mean you're gonna get away with killing everyone, but a sniper is a sniper. Next, there are three to four enemies that will guard/patrol the ledge on the mountain to the right of the sniper's nest. Pick them off one by one, running away after each kill. If you continue to kill from the same spot, the enemies will typically converge on your position and surround you. Once the ledge is clear, you can get to work on the rest. There are around 10-12 enemies guarding the building. Again, picking them off one by one was the best approach for me. I ended up getting lucky and once I had only five of them still living, they de-spawned. I doubt other's will be so lucky, but if there are still five guys wandering around, just keep picking one and running away. Taking on multiple enemies is not recommended. Once the area is clear, move inside the building and explore. Eventually you'll trigger a cutscene, and then you should get a checkpoint.
The next area is by far the worst. There are two points you have to plant explosives in. While neither points have many enemies guarding them, the area in general is infested with bad guys. Both of the points are also rather difficult to assault, and you have to destroy both before you get another checkpoint. There's a radio tower on a hill, and a second one behind a few buildings. You can pick which one you destroy first. I found the hill to be much more difficult, but I ended up succeeding after blowing up the one at the buildings first. The trick to this area is to kill, move to cover, kill, move to cover, etc. If you get hit, keep your head down and regen your health. Try to stay in a place where you can run from cover to cover, since the enemies will start spamming grenades and smoke. Use the smoke to your advantage to move and kill if it helps. This part will likely take a lot of trial and error. Just stay patient, learn from your mistakes, and keep trying. Some of this part is honestly just luck and situational awareness. Don't forget to keep an eye on your ammo, conserve when possible, and swap guns when necessary.
Once you've destroyed these two points, take a moment to be proud of yourself. The final part of this mission is pretty easy. You'll have to destroy two supply stocks. Luckily these are right next to a runway, and on that runway is a bomber. You can actually just run straight along the runway (from the direction the plane is facing) and if you stay on the right side of the plane (the plane's right, not your right) the enemies actually won't see you until you get in the plane and take off. If you're not an experienced flyer, you can still do this the normal way, but be warned, the bomber makes things so much easier. Once you've bombed both stockpiles you can easily fly back to your friend, parachute down, and the mission will end.
The third and final mission of this mini-campaign is essentially a firefight where you hold your position. You get several checkpoints in this level so don't get frustrated if you're dying a lot. You'll have four different types of enemies assault your position. Infantry are actually pretty easy to get rid of. There are two AA guns sitting in the area. Jump into one and wait for the soldiers to come up and you can easily blast them away. You'll fight some mortar vehicles as well. For these, there are two cannons facing the direction the enemy is coming from. Alternate between these two to blow up these vehicles. Some planes will also attack. Use the AA guns to quickly take these out (lead your shots to hit more often.) And finally some tanks will also attack. Use the cannons until the tanks get closer and then switch to an RPG (there's one lying on a box near the leftmost cannon.) Infantry will frequently come in with these vehicles too, so be prepared to jump back and hop into an AA gun to get rid of these guys. By any mean necessary, do not die when you and your friend start singing. This is the worst checkpoint to die at. You WILL have to listen to them play through the ENTIRE song again if you die. I would tell you to mute the game here, but your buddy will be making callouts for you that can help you stay alive. SO just suffer through that, and then survive one or two more waves, and then you're done.
A final cinematic will play and then the campaign will end. Congrats! Now just two more campaigns to beat, and the hardcore achievement is yours!