Generally speaking, yes, he could
Two-handed weapons require two hands, shields require one. If you have at least three, that means you can use both.
Shields do not have penalties for being in the “offhand”
That’s where they usually are, regardless of what weapon is otherwise wielded. Also, note that “offhand” doesn’t actually mean anything unless or until you choose to use Two-Weapon Fighting to gain an extra attack.
If you have more than two arms, you usually treat all hands except one as offhand. This again only matters for Multiweapon Fighting.
Tower Shields are really bad
Note that Tower Shields have lots of drawbacks: −2 to all attacks, huge Armor Check Penalty, tiny Max Dex, and they weigh a ton. They are rarely a good idea to use. I’d encourage your player to avoid it, not because it’s too powerful, but because it’s not powerful enough. A solid Heavy Shield gives half as much AC, but you ditch most of the problems you get with a Tower Shield. The Standard Action for Cover option is just not even remotely worth it.
And this is ignoring the fact that Aegides don’t get proficiency in Tower Shields, so he’ll have to multiclass or burn a feat to get it. That’s a big deal; feats should get you more than the option of using equipment that isn’t better than what you already could use.
The Aegis Customizations actually give two arms
As specified under the lesser extra arms customization, these options give a pair of arms, not just one. Note, however, that shields do not stack with one another, so that’s of limited benefit. Lesser extra arms can carry things, but not use them, so that is insufficient to actually benefit from any shield or weapon. Actually using a shield requires extra arms, minimum Aegis 5 and requiring a total of 3 points (2 points for extra arms and 1 point for lesser extra arms which is a prereq). That’s a pretty big chunk of a 5th-level Aegis’s available customization points.
Note that the fourth arm is difficult to use here. He could have a one-handed or light weapon in it, but unless he’s using Two-Weapon Fighting (and therefore has way more Dex than you’d want on someone who has a Max Dex of 2 on his shield, or is taking ludicrous penalties for TWF without the feat) he isn’t able to use that alongside his earthbreaker, so there wouldn’t be much point. Even then, it’s going to take another −2 penalty on attacks just for being in the extra arms; it takes greater extra arms (minimum 8th level, and 6 customization points) to avoid those penalties.
(If it comes up, TWF penalties are based on whether or not he has a Light weapon in the offhand; that his main weapon is two-handed does not affect these. Note he could do this without extra arms if he used Armor Spikes as his Light weapon. Thus, the probably-optimal way to use greater extra arms is Two-Weapon Fighting with a Two-Handed weapon, a Light weapon, and a shield, to minimize TWF penalties. But other than the shield, you could just use Armor Spikes to achieve the same thing, so this is not especially impressive.)
None of this is particularly overpowered
You have a 5th-level character with, as a sizeable portion of his accumulated class features, the ability to use a 2d6-damage weapon along with a shield. Note that a Rogue at this level could easily have 3d6 Sneak Attack damage, on top of a base weapon damage that could be 1d6 or more; sure, some things are immune to that (though quite a lot fewer in Pathfinder than in 3.5), and he has to do a little work to make sure it triggers (though Flanking is usually easy and to everyone’s benefit), but it’s still literally twice as much damage.
He does have more AC, plus the DR of the Juggernaut Armor, but those are his class features, and are a big part of the reason why he’s doing half as much damage, has half as many base skill points for a much smaller list, and so on. And if he actually goes for the Tower Shield, his huge penalties to attack and many skill checks are even bigger problems. Note that for the same penalty to attack, your typical Rogue could double his damage by using Two-Weapon Fighting, so now his damage is quadruple what the Aegis gets.
The Aegis gets to be really, really difficult to hurt with physical attacks. That’s his schtick, that’s what the class is good at. It’s a pretty narrow schtick and I worry that he’ll find that smart enemies just ignore him since he’s a pain to kill but has low damage output. Maybe he’ll invest other customizations towards improving that, but Aegides aren’t exactly amazing in that department; he’s never actually going to catch up with the Rogue who gets his dual-wielding full-sneak-attack. That’s the Rogue’s thing; this is the Aegis’s. I recommend against punishing players for being good at what they’re supposed to be good at.
You cannot benefit from two shields at once. They both provide Shield bonuses to AC, which do not stack with one another. Moveover, as you note, those bonuses become void the moment you actually attack: rather than give a bonus to AC, the bucklers start giving you penalties to attack.
Your attack routine therefore looks like this:
Main: +1 (BAB) +3 (Str) −2 (TWF) −1 (Buckler) = +1
Offhand: +1 (BAB) +3 (Str) −2 (TWF) −1 (Buckler) = +1
AC: 10 + Armor +3 (Dex) +0 (Shield)
When not attacking, your AC is 10 + Armor + 3 (Dex) + 1 (Shield).
A +1 attack bonus is very bad, even at level 1. Do not expect to reliably hit much of anything with that attack bonus; a typical low-AC target is still looking at AC 14 or so. You’d only have a 64% chance to hit even once against that target. Meanwhile the barbarian’s looking at around a +7 attack bonus, for a 70% chance to hit, for a whole lot more damage. Your odds of hitting with both attacks, 16%, is so low that you might as well not even have Two-Weapon Fighting.
That said, you really do not need to take these penalties to play the character you want to play.
Armor in Pathfinder is fairly heavily abstracted; one type of armor can actually cover quite a lot of fairly-different sets of protective gear, all of which is simply abstracted into a single armor bonus, a single armor check penalty, a single maximum Dexterity bonus to AC, a single weight, a single cost, and so on. Thus, while one person’s chain shirt might be literally that, a shirt made from chain links, I see no reason why you couldn’t describe yours differently: lighter protection on the body, perhaps, but heavier arm bracers, used to deflect blows.
Thus, your description becomes just that: description. There is no need to fall into 3.x’s “you need the feat [item] for that” fallacy. Simply describe your character the way you wish to describe him or her, and use the items that most effectively represent that description mechanically.
I cannot vouch for how nit-picky your PFS DM will be on this, however. Some DMs are incredibly nit-picky about completely unnecessary things, and in PFS they have some institutional backing for that. I personally dislike the PFS immensely, largely for this reason. Normally, I’d say if a DM gave you a hard time about this, that is a DM you’re better off not playing with, but here you may not find one who will. Ultimately, you may not be able to effectively make the character you want to make, at least no without taking absolutely pointless penalties (as with the paired bucklers).
Best Answer
There are a few ways, but bear in mind that it's much easier to increase your attack bonus than simply remove the penalty, with greater benefits in the long run (such as unlocking other feats):
There is a third party (Kobold Press) feat that does exactly what you are looking for: