Why are non-spontaneous reactions driven in the presence of energy?
You are incorrect in your assumption that the addition of a power source makes both the forward and the reverse redox reactions more spontaneous. An electrochemical cell operates on the same principle as a battery. On the anode of the cell, you will find a compound that is being oxidized, or losing electrons. These electrons then flow through some sort of a wire to the cathode, where the electrons are being used to reduce another compound. The electricity will flow in the "spontaneous" direction, in other words, from the compound that prefers to be oxidized to the compound that prefers to be reduced. The "voltage" across an electrochemical cell is determined by adding up how badly the reducing agent wants to get oxidized ($E°_{ox} = -E°_{red}$) and how badly the oxidizing agent want to be reduced ($E°_{red}$), and is a measure of how much the reaction wants to occur. 

Now, lets try adding a battery in between the anode and the cathode. If the battery is added such that its anode points towards the cell's cathode, the battery will add to the voltage of the cell, causing the effective voltage for the reaction to increase. On the other hand, if the battery is placed such that its anode points towards the cell's anode, it will be creating a voltage in the other direction. When this happens, it becomes a question of whichever voltage is stronger. If the battery has a stronger voltage than the cell, electrons will flow in the opposite direction. This is what occurs in electrolysis, causing the originally non spontaneous process to become spontaneous. This results in charging the electrochemical cell. On the other hand, if the battery has a smaller voltage than the cell, the electrons will flow in the original direction - charging the battery as they pass through it.

While a power source can be used to either make a spontaneous process more spontaneous or to reverse the flow of electrons, it cannot do both at the same time.