StepStick Protectors
To Protect, or Not to Protect

From the web site you reference:


  The SilentStepStick Protector is an add-on module for StepStick and
  Pololu A4988 compatible stepper motor drivers. The board provides
  flyback diodes (freewheeling diodes) for the motor outputs, so that
  they are also protected against induction voltages in the unpowered
  state of the driver.
  
  The v2 protector also contains a diode from the logic voltage (anode)
  to the motor voltage (cathode), so that the power supply order of all
  SilentStepSticks is always guaranteed.


Since I assume you've read this, please indulge some explanation of the electrical engineering.

Motors

Every motor has a part called the "rotor" that rotates, and a part called the "stator" that is stationary.  

In the stepper motors usually used in 3D printers, the rotor is a permanent magnet and the stator consists of coils of wire.  In normal operation, the current flowing through the coils creates a magnetic field.  The rotor aligns itself with the field of the stator.  To move, the stator field is shifted by changing the current in the coils, and the magnetic force causes the rotor to rotate to align the fields.

Generated Voltages

As Faraday discovered, a changing magnetic field induces a voltage in a wire.  There are two ways that the magnetic field changes.  The first is that the current changes or stops.  That introduces little high voltage blips into the motor coils.  The motor drivers are well prepared to handle this blips.  The second is when the rotor is forced to turn by an outside force, such as sliding the bed.  As these voltages are not related to action the motor driver is causing, some of the techniques used to handle the voltage spikes may not be as robust.

The protection boards you point to consist of diodes to limit the magnitude of the voltages that the coil can produce.  This is redundant with the diodes and protections built into the driver chips, but it may offer stronger protection.

Benefits?

In this responder's opinion, you are unlikely to see a great benefit from these boards, but perhaps there are vulnerabilities I am unaware of.  You are also unlikely to do any harm.

You asked specifically if, with the protectors installed, you could unplug the motor while it is passing current.  The protectors may help with that, or they may not.  Interrupt the connection creates a spark, and arcing through the air, that makes many high-frequency components which may not (or may) be effectively clamped by the diodes.  For best results, you could add the protectors, and also avoid interrupting the motor current while powered.