S Tier
Scout - The ultimate and passives of this class are a little underwhelming, but Combat Recon is one of if not the best abilities in the entire game. Being able to gain extra Advantage each turn unlocks some of your ADV-heavy classes such as the Heavy, and is critical for certain team comps.
Heavy - taunt is one of the best mechanics in the game, and this class has a large AoE version. Incredible when paired with the Retaliation stacks from Unstoppable and hiding behind large cover.
A Tier
Gunslinger - can fire off a multitude of attacks in a single turn, giving this spec extremely high dps output and the best for dealing with a large group of weak enemies (ignoring grenade use).
Soldier - The Rocket Strike ultimate is the best in the game. It costs a whopping 10 advantage, but is well worth the investment and can be upgraded to one shot most normal enemies. Great for taking down reinforcements. Brutal Strike's Dazing effect is also very powerful.
B Tier
Assault - Sweeping Salvo is a great damage in a large area with the added bonus of a debuff. Underwhelming outside of this ultimate.
Sharpshooter - Snipers are generally very strong in XCOM games, but with all classes being able to wield one, the Sharpshooter loses its originality a tad. However, Overwatch remains incredibly strong, and this class has great OW boosters, but is lacking in other areas.
Medic - the ability to heal is decent, especially on higher difficulties where taking damage is harder to avoid, but the real strength is the Combat Stim, buffing your allies damage and survivability.
C Tier
Scoundrel - No classes are truly bad in Zero Company, but the Scoundrel requires more set up to be useful making it a bit tricky to pull off.