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.

Bucklers provide shield bonuses to AC. If you have two, you have two shield bonuses to AC, and as typed bonuses they do not stack—you use the higher one.

Any penalties, though, do stack. That means the attack penalties, the armor check penalties, and so on. The arcane spell failure would stack as well, though if both are mithral, 0% + 0% = 0%, so you would still have none.

For the purposes of other bonuses from special properties, though, I see no reason you couldn’t use two. But I would definitely ask the DM to make sure he or she is OK with it because it is kind of unusual.