Ugnaught Demolitionist x1
Mouse Droid x2
Rodian Brute x8
R7 Astromech Droid x1
Caamasi Noble x1
Ugnaught Droid Destroyer x2
Treadwell Droid x1
Twi'lek Black Sun Vigo x1
Gamorrean Bodyguard x1
IG-86 Assassin Droid x3
Kel Dor Bounty Hunter x1
Jawa Scout x1
Lobot
Wicket
Salacious Crumb

I don't know if I count, since I was part of the original conversation, but in my opinion, the only piece that is completely essential to every squad is something with the ability to blow up doors. Unless you've got something in-faction, like Spaarti Clone Troopers or Juggernaught War Droids, Ugnaught Demolitionists are best. In a 150 or 200 point game you want 3 preferably, and 2 at a bare minimum.

After that, Mouse Droids are essential if you need to transmit Commander Effects, and they are also helpful as blockers.

Rodian Brutes are essential if you are running a swap squad, like with the new Revan. If you do need them, you need at least 3-4.

I think that's the really essential pieces - with Ugnaughts, Mouse Droids, Rodian Brutes, and say a handful of v-set Old Republic or Sith pieces, you can make a tier 1 squad.

After that, you get into Lobot, who can bring in stuff like Salacious/Gamorrean Bodyguard against the Lancer, a Rakghoul against nom bombs, an R7 for double override, or a jawa scavenger if you're playing Seps.

LeftiesWillRule list looks pretty good, except I think you need more than one Ugnaught, and I'd probably chuck in a Rakghoul (Celeste Morne is pretty cool, and it helps you play her) and a Czerka Scientist, which is very helpful for some builds. I'd probably suggest up to 4 Camaasis as a potential Lobot option.

Have a new player print off an Atris card and then pick up an Old Republic Jedi Guardian and a handful of Echani Handmaidens for really cheap and you have yourself a good start to a strong squad that even a noob could use with some success.

I would go:

Ugnaught Demolitionist (Minimum 3) - Self explanitory
Lobot and/or Lobot, CLO - Good override character. Uni Lobot can help tailor a squad and CLO's Recon/Reinforcement combos are nice.
Dark Hellion on Swoop Bike (only 1) - Good, cheap and fast unit for gambit scoring/opening doors without override
Human Bodyguard (Minimum 2) - Bodyguards are always nice. One Houjix would be good as well.
Kel Dor Bounty Hunter (Minimum 2) - Good, Solid attacker that helps deter melee beatsticks.
Czerka Scientist (only 1) - Can't really go wring with Twin or Jolt.
IG-86 Assassin Droid (Minimum 1) - Good shooter that can help round out squads until the collection bulks up. Twi'lek Black Sun Vigos work as well
R7 Astromech Droid (minimum 1) - Cheap Override

Everything else, while beneficial, is not overly necessary for a beginning collection. I do not consider Mouse Droids essential, simply because for a beginner, they can all too easily become a crutch (like Mas Amedda and Boba, BH). If a player can learn to play effectively without them, they then become a nice bonus rather than a requirement.

In my experience, I've never needed more than one Ugnaught Demolitionist except for one game against a squad of 4 Elite Rebel Commandos. I get more use out of a Rodian Brute than a second Ugnaught since I'm usually able to keep him safe until he can blow up the one important door and then Override isn't important (except maybe on Throne Room). Only a jerk would play 4 override pieces against a newbie, so one should be fine to get him started.

Good lists above.
I would reiterate that Jarael is very useful.
Lando DS is still a solid option for a cheap fringe shooter.

With the R7 getting T3-M4 is wasteful. Both provide over ride but the R7 had more hp, immunity to extra damage from Ion Weapons, and is not unique. The R7 also costs 9 points less (or even more) which is enough room for some kind of shooter to make up for T3-M4's limited attack potential.

I do think that the Dark Times bounty hunters are a pretty good value that shouldn't be hard to find.

Now my thoughts are:
1 Mouse Droid (although 2 to 4 should be more than enough unless you start spamming them)
2 Uggies (there are times when one isn't going to be safe enough)
4 Rodian Brutes (these are fill out fodder)
1 Caamasi Noble
Lobot
R7

There are others but those are the pieces that seem to show up across the widest array of squads.

Well, let's see what's in my reinforcements box:

Ugnaught demolitionists (you really only need one for the card, then just use the ugnaught figures from IE)
Mouse droids
Rodian brutes
Gran raiders (stealth swappers are often very useful)
Anamin scout (no one has mentioned him, he's one of the best swap pieces in the game)
Nobles
Twi'lek bodyguards
Human bodyguard
Gamorrean bodyguard
R7
10-point It's a Trap! droid
Targeting droid
Kel-Dor bounty hunter
Wicket
Salacious Crumb
IG-86
Czerka scientist