Take all hail options when they appear. If the answer "Save them or die trying" comes up, that's the jackpot, take it.
You will receive a quest for the next sector where you must wait until the dice read 5 or 8... Just kidding, you have to outlast a rock ship in a solar flare. Good luck.
The easiest way to do this (I cant remember who im quoting but thank you) is to move all of your essential crew to the medbay, place any rock characters in vital areas (o2 or engines) and wait. Focus solely on keeping the ship alive and making the jump away after they do.
They should appear in a system directly adjacent to where you were and you will gain their support, a weapon (I think) and the ship.
The rock ship type a has a blue option that makes picking up the stasis pod happen 100% of the time. Well worth it.
I used to be capable of having children. Then I took an arrow in the knee... Xbox tag - Dtothe3
I have the same ships left to acquire as the TC does, after getting the Stealth Fighter last night, and by God the RNG is a cruel mistress. In my last ~30 playthroughs, including some where I deliberately took hostile-heavy paths for the sole purpose of searching for the Mantis homeworlds, I've seen them exactly once, on a playthrough where I hadn't had an opportunity to acquire a Mantis and was playing on a ship that did not start with one. Though there seems to be some ambiguity regarding the Mantis crewmate's necessity on the wiki thing, it wouldn't have mattered anyway because I never saw the ship. Which I've seen exactly once, and it was literally on my third or fourth game. 60 or 70 games later... no dice.
I gather that the Unidentified Cruiser, in particular, necessitates the planets in many different star systems all aligning at once to make a giant heart with your name inscribed in the center.