Javik the Prothean is a DLC only squad member that's available in the "From Ashes" DLC:
Access to From Ashes is included free with all copies of the Mass Effect 3 N7 Collector's Edition and the Digital Deluxe Edition. It is available for purchase separately for 800 Microsoft Points, 800 BioWare Points, or $9.99 USD on PlayStation Network.
And as @LessPop mentioned, the events in previous Mass Effect games will mean that there will be some squad members that are completely unavailable in the game:
Ashley/Kaidan depending on choices in ME1, and Garrus and Tali depending on their survival of the suicide mission in ME2.
Other than that, the only way to 'miss' a squad member is to follow certain decisions in ME3 which results in their death. There's no way to avoid recruiting any of the non-DLC squad members in ME3, they all show up as a part of the main quest.
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
Best Answer
In the same way that they don't run out of ammo, they are not affected by weight restrictions
See here