LIAM – Liam is the first you’ll get, and he’s unlocked once you reach Habitat 7, which is simply part of the story.
CORA– Cora is also obtained on Habitat 7, though you’ll meet up with her later on, about halfway through the first mission. Just follow along the story to get her.
VETRA – Vetra joins your crew right when you leave Nexus. She helps get your ship cleared then jumps right on board with you.
DRACK- You'll run into this Krogan on Eos, while doing one the first story mission checking out the 3 monoliths, when you run into him a short cut scene will let you talk to him for awhile. You will then run into him again while going to setup your first outpost he will be at the site fending off some Kett, after a short cut scene he will then join you.
PEEBEE – Peebee is also found on Eos. You’ll first encounter her when you start interacting with the Monoliths on the planet. After exploring your first vault she’ll decide to join you to learn even more.
JAAL – Jaal is the new race of angara and you’ll meet Jaal during the main mission “A Trail of Hope.” to the Angarnian people your an alien and treats you like one, after meeting a few people there and various cut scenes Jaal will join you to learn more about your kind.
So in short you gain every squad member through natural progression. 5/6 on EOS and last one on AYA.