The special glowing beacon, when spawning in mountains or ice, can spawn inside of pre-damaged/cracked tiles. This means that at the beginning of every battle, just look at every square. You may see a beacon glowing right from the start! You just have to know what to look for. See section 2 for many examples of beacons in mountains and section 4 for an example of a beacon in ice so you know what exactly to look for.
In fact, the vast majority of beacons you will probably encounter are found in these instances of pre-damaged/cracked mountain and ice locations!!!

1A. Set the game difficulty to EASY mode. You are going to be wasting a lot of time and firepower cracking mountains, and there's no way Normal or Hard will give you the luxury of free actions to break open mountains. Not only do you need to crack dozens if not hundreds of mountains to find what you are looking for, but once you crack it, you need to then destroy it and move a unit to pick up a drop.

1B. Bring Mechs with lots of firepower, and AOE firepower as well. Science types that deal no direct damage are of no help here. Mechs like the Mirror Mech that can hit two distant mountains at the same time or the Siege Mech that deals damage all around the target square are your best friends for this hunt. Flame Leviathans are probably one of the worst squads, as none of them deal direct square damage!

1C. Once you have your squad, it's mountain cracking time. Control the Vek with as few moves and effort spent as possible (this is why you want to be on easy mode). Then spend all extra actions you have left over using your Mechs to attack mountains wherever you see them. Once you crack a mountain (1 damage) you should be able to see the glow right away. If you don't, move on to the next mountain. Be prepared for a long grind, I spent at least 5, 10, or maybe even 15 timelines focused just on cracking mountains, staying alive so I didn't get a game over, and not much else.

2A. The glow you are looking for in a mountain, note the bottom left mountain has a yellow dot inside.
http://cs2bus.com/sharedfiles/filedetails/?id=1318139727
Also, that same glow on an ice mountain, which is even harder to see because the yellow blends in with the white.
http://cs2bus.com/sharedfiles/filedetails/?id=1318866402
2B. Pilot makes a remark, and you see a small beacon on the square now.
http://cs2bus.com/sharedfiles/filedetails/?id=1318139853
2C. Move a Mech to pick up the small beacon.
http://cs2bus.com/sharedfiles/filedetails/?id=1318139995
2D. Special FTL pod will arrive. Thankfully you only need to protect it, you don't have to hustle over and actually pick it up during the mission.
http://cs2bus.com/sharedfiles/filedetails/?id=1318140092
2E. Opening the special FTL themed pod...
http://cs2bus.com/sharedfiles/filedetails/?id=1318140140
2F. Earning the achievement.
http://cs2bus.com/sharedfiles/filedetails/?id=1318140191

3A. There appears to be three total special pilots. From what I have gathered, they are each tied to a specific zone. Mantis in Grassland, Rockman in Acidland, and Zoltan in Desertland. Striked out statement proven to be false. You only need one to unlock the achievement.

3B. Each Pilot has very unique stats, if I ever acquire them all maybe I'll update the guide to show them all.

3C. As far as I can tell, there is no hidden pilot in the final zone, nor in the Iceland zone. I am very much NOT 100% certain about the Iceland zone not having a pilot though, I just have not found any evidence yet to support that zone having a pilot. Striked out statement proven to be false.

3D. Happy hunting, hope this guide was helpful! If you found this guide helpful, please give a good rating so others can find it with high visibility!

4A. Pods can be found in the Ice (Pinnacle) Zone. In fact they can be found here buried under the ice as well. So get cracking with those AOE attacks!

http://cs2bus.com/sharedfiles/filedetails/?id=1318452608

http://cs2bus.com/sharedfiles/filedetails/?id=1318452676

Credit to: Cyriel ᛭ᚲᛃᚱᛁᛖᛚ᛭ for the Ice Beacon Screenshots.

4B. As noted above, beacons are confirmed to be found inside of mountain squares and also potentially now in frozen water/ice squares.

4C. Specific special FTL aliens are not linked to a specific zone. Here is the Mantis being found in the Acid/Detrius zone.

http://cs2bus.com/sharedfiles/filedetails/?id=1318492304

4D. Special beacons can be found on any difficulty mode, including hard. But good luck having the spare time and resources to spend searching and acquiring them on hard!

4E. Special beacons, when spawning in mountains or ice, can spawn inside undamaged tiles or pre-damaged tiles. This means that at the beginning of every battle, just look at every square. You may see a beacon glowing right from the start! You just have to know what to look for. See section 2.

As title suggests, take this section's information with a hint of skeptisms. It's only been proven through player observation and not 100% verified.

So it seems there are several locations that it is impossible for the special beacon to appear in, and hopefully knowing this will save you some time in your hunt!

1. The very first mission. So don't think you can abandon a timeline over and over until it shows up immediately.
2. The first island, whichever island you pick. Same vein as #1, it seem that at the earliest you can find a special beacon is from island 2 and onward. Could save a lot of trouble cracking mountains right from the start.
3. Any mission that already has a time pod that appears. The appearance of the normal time pod seems to make the extra one never spawn.
4. Island boss battle. Never ever seen a pre-cracked terrain in any boss battle, and never got a beacon from breaking mountains either.

Again though, these are all unconfirmed reports, but if they are true, it should help somewhat in narrowing your search. Good luck!

Using the above strategies and tactics, I was able to acquire every FTL alien pilot with just a few hours dedicated effort. So here are all the portraits of the secret pilots as well as their stats!