Mass-effect – significance of the red star in the Reputation bar of the Squad screen

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

No, there is no easily-accessible indicator of squad health. I sent my squad mates head first into a suicidal combat and watched them take damage, but it was not reflected in their tiny HUD pictures.

I have seen the portraits flash briefly, but I don't know what caused that to happen.

There's likely no way of showing this information on console versions of ME3, but it's probable that someone will mod this feature into the PC version.