Who are all the squad members in Mass Effect 3?

Your squad in ME3 consists of:

The Virmire Survivor, which means one of Ashley or Kaidan, as well as the return of Garrus, Tali, and Liara. In addition, two new members join the squad, James, and a newly corporeal EDI. If you own the From Ashes DLC, there is also Javik, the Prothean.

As for 'classes', the squad members don't have classes in the sense that Shepard does. Instead they each have their own unique set of 4 powers. Two are shared with one of the available Shepard classes, and two are unique to them, but can be unlocked as Bonus Powers through conversation.

Specifically, the lineup of powers for each looks something like this:

Ashley Williams

  Disruptor Ammo     Concussive Shot       Inferno Grenade   Marksman  

Kaidan Alenko

  Reave              Barrier               Overload          Cryo Blast

James Vega

  Incendiary Ammo    Carnage               Fortification     Frag Grenade

Garrus Vakarian

 Concussive Shot     Armor-Piercing Ammo   Overload          Proximity Mine

Liara T'Soni

  Singularity        Warp                  Stasis            Warp Ammo

Tali'Zorah vas Normandy

 Defense Drone       Energy Drain          Combat Drone      Sabotage

EDI

 Incinerate          Overload              Defense Matrix    Decoy

Javik

 Dark Channel       Lift Grenade           Pull              Slam

Searching around on Mass Effect Wiki The constant squad members throughout the game are as follows:

• Virmire SurvivorAsh or Kaidan
• Liara T'Soni
• Tali'Zorah vas Normandy
• EDI
• James Vega
• Garrus Vakarian
• Javik (DLC)

Excluding future DLCs, here is your entire list of party members that are currently available:

• The Virmire survivor (Kaidan/Ashley)
• Tali'zorah vas Normandy
• Liara T'Soni
• Garrus Vakarian
• James Vega
• Javic The Prothean (DLC only)
• EDI

Source