The commonest answer to these 3 follow-up questions is: 'someone has blue eyes' must be common knowledge, for all the blue-eyed people to leave. But why? Don't you only need know your own eye color to leave?
This is clearest where $n=1$. You don't care whether it's common knowledge that someone has blue eyes. Only you need to know that someone has blue eyes, and given that you don't see anyone else, you will deduce that 'someone' must be you.
The case where $n=2$ is similar. Now you do care whether the other blue-eyed person knows that there is someone with blue eyes, but you don't care that they know that you know the same fact (hence you don't care whether it's common knowledge). As long as they know, and they don't leave the first night, then you will deduce that they also see someone with blue eyes, which must be you.
Suppose that the guru told each of you separately that there is someone with blue eyes, but she only told you that she told the other guy this fact. Then you would be able to leave on the second night, whereas the other guy would not.
So why does common knowledge matter for being able to leave?
Also, common knowledge doesn't seem to matter because it is defined as having an infinite series of "knows that" propositions, but here n=100 is finite.
ETA: Common knowledge is used as an argument for why the guru's imparted information matters, but if common knowledge doesn't matter, then the question remains: why does the guru's imparted information matter (i.e. what new information does it contain)?.
I would guess that it contains no new information and the inductive solution is in fact invalid, and that no one leaves the island.
Best Answer
I give a quite detailed answer at the linked question, that tries to explain in detail how each event changes the state of (mental) affairs on the island, and how much of "common knowledge" is really required to eventually cause the individuals concerned to leave. In my notation there with $C$ for "it is common knowledge that" and $E$ for "everybody knows that", common knowledge $C(P)$ means $\forall k\in\Bbb N:E^k(P)$; of the information $C(n>0)$ caused by the declaration, it is the instance $E^{100}(n>0)$ that ultimately gets things moving.