Hybridization, sharing and overlapping of electrons takes place only for covalent and co-ordinate compounds. Carbon exhibits the same due to catenation and tetravalency.
There is no quantum mechanical explanation of the octet rule which is full proof because it isn't, as a matter of fact, completely true. However bonding and overlapping occurs due to the quantum numbers.
For instance, carbon has the configuration which looks like this:
In its excited state (when given energy) one electron from the 2s orbital reaches the p orbital, thereby leaving 4 electron spaces for bonding purposes and complying to Hund's rule. The excited configuration looks like this:
And the above develops varying kinds of hybridizations, the most common ones being sp3, sp2 and sp.
However the ionic bond formation is different as compared to the above process. Here, the electrons undergo complete sharing of electrons. Take the example of Sodium chloride. A common question which arises is that if Sodium chloride is formed due to formation of Sodium and Chlorine ions separately, why do both the ions stay together? Ideally Sodium and Chlorine both have achieved their octets, by losing and gaining an electron respectively.
As you can see in the above diagram, the two ions stay in close contact due to attraction of the opposite charges developed on them due to loss and gain of electrons respectively. This is mainly due to electrostatic attraction developed between the ions. No overlapping of any sort occurs in an electrovalent compound.