Physiology of hyperpolarization
I don't get to say this on very many occasions with questions like this...

Your textbook is wrong.

A typical potassium reversal potential in a cell is ~-90 mV. Hyperpolarization through voltage-gated potassium channels can never go more negative than that reversal potential.

An experimenter with access to the cell through a patch clamp electrode could possibly set the voltage more negative than potassium reversal, in which case indeed, potassium would flow against it's concentration gradient into the cell due to the electrical potential. It is not possible to reach those very negative potentials using potassium channels alone.

For a typical cell, the resting membrane potential is more like -70 mV. This resting potential is due to the "leak conductance", which does include potassium but also includes other ions. The ratio of sodium to potassium permeability is typically around 1:20; potassium dominates but you can't just ignore the other ions. A more accurate replacement statement for your textbook would be:


  The leakage conductances tend to drive the membrane potential from hyperpolarized state to the resting state as the net positive flow of ions is inward.


Even at rest, you will always find a slow potassium leak out of the cell. The reason that the cell is at equilibrium at rest is because this leak out of the cell is perfectly balanced by net positive charge flowing in to the cell; this is mostly sodium but can be other ions like calcium as well (note also that that chloride leaving the cell is also a net positive charge in).

Some of the sodium leak current is through specific sodium leak channels, but co-transporters that use the sodium concentration gradient also contribute. 



Ren, D. (2011). Sodium leak channels in neuronal excitability and rhythmic behaviors. Neuron, 72(6), 899-911.