You have a constraint system. For every surface-element the tangent component of the electrical field strength must be zero (else the surface charge would start to move).
Consider a perpendicular edge of a cylindrical configuration as shown in the following picture.
Assume that the surface charge distribution would be uniform (constant charge per area).
There we have small surface charge elements $\sigma dA_1$ and $\sigma
dA_2$ with $dA_1=dA_2$ generating a force on a small surface charge element $\sigma
dA_3$ very close to the edge.
Thereby, $\sigma dA_1$ and $\sigma dA_2$ are just samples of the field generating surface charge elements over which we have to integrate. Nevertheless, we can discuss the main effect exemplary with their help.
Both $\sigma dA_1$ and $\sigma dA_2$ have the same distance from $\sigma dA_3$ so that the absolute value of their field strength at the position of $\sigma dA_3$ is the same.
The tangent component $\vec F_{2\parallel}$ of the force $\vec F_2$ caused by $\sigma dA_2$ on $\sigma
dA_3$ is small since $\vec F_2$ is directed almost normal to the surface.
On the other side the force $\vec F_1$ caused by $\sigma dA_1$ on $\sigma dA_3$ directly has tangent direction ($\vec F_1 = \vec F_{1\parallel}$).
Therefore, the sum of tangent forces $\vec F_{1\parallel}+\vec F_{2\parallel}$ on $\sigma dA_3$ will point towards the edge. In principle his holds also for the other charge elements contributing to the electrical field at the location of $\sigma dA_3$. Consequently, the surface charge $\sigma dA_3$ will move towards the edge until there is so much charge in the edge that for all surface charge elements the tangent component of the field strength is zero. ("The edge repells further charge.")
Eventually, at edges the surface charge density becomes infinite. Instead of a finite surface density you have a finite line charge density in the edge.
The effect at curved surfaces is similar but not so drastic.
The statement of the book that only spheres admit uniform charge distribution is only right if you restrict your considerations to bounded conductors and fast enough decaying fields at infinity.
If you admit infinite conductors then you also have uniform surface charge on a circular cylinder.
If you admitt outer charge distributions then you can adjust these outer charges such that the charge distribution on a considered smooth surface is uniform. Thereby, no restrictions are made on the shape of the surface.
Surface charge refers to a thin layer of charge at a surface. For example, an ideal charged conducting sphere would have a layer of charge on the surface whose thickness would be infinitesimal. It would be represented by a delta function.
In the case of an idealized non-conducting uniformly charged sphere whose boundary is discreet, the charge density simply goes from a finite value to zero. No surface charge.
If we relax the restriction to ideal spheres, the conducting sphere has a charge density that spikes at the surface, while for a "uniformly charged" sphere, the charge density would drop smoothly to zero without a spike. (But it might have some structure depending on what model you use for the material and the electrons.)
Best Answer
If electrons obeyed classical mechanics, they would rearrange in a new configuration in order to maximize the distance between them. They would not stay still because of thermal motion, as pointed out by CuriousOne, but on average they will still maximize this distance.
However, electrons don't obey classical mechanics, but quantum mechanics. The behavior of electrons in a conductor is more like that of waves (a plane wave in the free electron model, which is the crudest approximation). So the electron are delocalized on the whole surface of the conductor and it is probably more meaningful to talk about a single charge density function $\rho(\mathbf{r})$ rather than talking about individual electrons.