Hexagonal unit cells
The cubic primitive, cubic body-centered and the cubic face-centered are just the three possible Bravais lattices of the cubic crystal system.

In total, we know seven crystal systems, namely (in order of increasing symmetry) triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic.

Among these, the triclinic crystal system has the lowest symmetry ($a \ne b \ne c;\ \alpha, \beta, \gamma \ne 90°$), the cubic crystal system has the highest symmetry ($a = b = c;\ \alpha = \beta = \gamma = 90°$).

A careful look at illustrations of the hexagonal closest packing ($7+3+7$ atoms in $ABA$ layers) in a hexagonal prism and a bit of thinking gives the number of atoms:


$3$ atoms in the $B$ layer are not shared by any other prism. They fully count 
as $\mathbf{3}$.
$2$ atoms in the centre of the top and bottom $A$ layer are each shared by any other prism (above and below). Each of them counts $\frac{1}{2}$. So there's another $2 \times\frac{1}{2}= \mathbf{1}$ atom.
Top and bottom $A$ layers each have $6$ atoms on corner positions. Each atoms is shared by three neighbors in the the same layer and with another hexagonal prism on top or below. So they count as $2\times6\times\frac{1}{3}\times\frac{1}{2}=\mathbf{2}$ atoms. 


The total count thus is $\mathbf{6}$.

These diagrams of the hexagonal closest packing, do not represent the the unit cell as the most simple unit from which the crystal is build by symmetry operations. 

Cutting the hexagonal prism (along the axis through the atoms centering the top and bottom faces) into three pieces gives identical rhombohedra. These can be transferred into each other by rotations along the axis described above.