Obviously this won't be the squad, but just for fun...
Human Male Soldier - North American. Gunnery Sgt. Ellison (trained Shepard and Williams), a grizzled Staff NCO and former Drill Instructor at Macapá. Rather than the relative n00b who is less experienced than the protagonist and looks up to him/her like Kaidan or Vega, he has spent over twenty years in the military and fought in both the Skyllian Blitz and Reaper War, is a bit of a hard ass, and not afraid to let the protagonist or other officers know when they're wrong. He should be the stereotypical lifer and someone you couldn't imagine existing outside the military. Think someone similar to Quaritch from Avatar, except enlisted, or Sergeant Major Plumley.
Human Female Infiltrator - Lieutenant. Born in Lowell City on Mars, the child of Eastern European & Iranian immigrants. Evacuated from Mars to the Citadel along with other civilians after Earth was attacked, she later enlisted in the Alliance from the Citadel's docking bay refugee camp. Was wounded and received a battlefield commission for heroism at Firebase Dagger/Ontarom. Pulled strings to get two surviving kid brothers aboard the ark. Not a lifer and plans to leave the military for medical school, but not before helping to find a new home where her family can be safe.
Asari Vanguard - 350 year old Asari commando, from the colony of Lusia. Took part in the defense of the Cyone after the Reaper attack. Sarcastic and has a dry sense of humor. Countless close calls with death have left her with a strong belief in living life to its fullest, and she has embraced the Asari equivalent of an Epicurean philosophy.
Quarian Male Engineer - An artificial intelligence expert who was exiled from the Migrant Fleet for conducting illegal A.I. research, over the objections of Admiral Xen. Lived on Omega during his exile. Later recruited by the Shadow Broker (Yahg) as part of his cyber intelligence network. He remained working in that capacity after Liara seized control, and was one of the Shadow Broker's key assets during the Reaper War. Doesn't know the Shadow Broker's identity but has suspicions of a regime change based on how the organization operated following the events of LotSB. Vital in cracking Reaper code that was passed on to various Council intelligence services, helping to thwart an attack on the Salarian colony of Mannovai.
Batarian Male Sentinel - Born on Khar'Shan. Part of the elite Batarian Special Intervention Unit, he was recruited for a black ops mission that later evolved into the Skyllian Blitz. While serving as an adviser to the mercenary and pirate proxies attacking human colonies, he was wounded and captured during the failed assault on Elysium. He spent several years in Alliance military detention centers, stubbornly refusing to provide actionable intelligence while under interrogation. In the chaos and confusion of the Reaper invasion, he was able to escape from his Alliance holding cell. Stranded on a human world and a firm believer in the old maxim that an enemy of an enemy is a friend, he joined with a band of human civilian refugees in the wilderness and forged them into a hardened resistance cell. Later he was aboard one of the last ships carrying refugees off planet, after being seriously wounded. While he maintains loyalty to the former Hegemony, he developed a grudging respect for humans after spending months fighting beside them.
Male Yahg Soldier- Captured by Batarian slavers who sold him to a Terminus warlord fond of watching slaves fight and kill each other for sport. After several years as a prized pit fighter, he led a slave revolt that murdered his master. Escaped in his master's ship and went pirate. Founded a small colony of runaway slaves turned pirates on a derelict space station where he ruled as their 'king.' Terrorized the shipping lanes of the Terminus for years before losing it all during the Reaper War. Learned of the ark project during a card game with a drunken Salarian engineer, and managed to get smuggled aboard as a stowaway in a cargo container through a bribe. Discovered and imprisoned during the ark's flight. Selfish and aggressive demeanor. He will only serve the protagonist if dominance is first established. (besting him in combat or a high intimidation check)
Turian Female Adept - Spacer, born on the Citadel. A member of the Turian cabals, a battle scarred-veteran of the Reaper War and a lifer. She was stationed on Palaven when the Reapers attacked and spent weeks on the front lines. Severely wounded during the fighting and evacuated to the Citadel for medical treatment, where a lost arm was replaced with a prosthetic. Her mate and child remained on Palaven and were killed during the fighting. Suffering from survivor's guilt.