When the particle with charge q reaches the center of the original square, it is, as stated in the problem, momentarily at rest. Is the particle at equilibrium at that moment?

NO

Explanation:

The answer is NO because when the particle reaches the center of square there is a net force acting on the particle dew to various other charges and this net force gives acceleration to the particle. Moreover, For particle or object to be in equilibrium the net force acting on it should be zero and hence no acceleration. Although velocity can be zero or non zero at equilibrium state.

The context is missing here, but ill try to explain a general case.

Something is in equilibrium if it is in a valley of the potential energy, this is because things in life try to be in the minimal energy state possible. Think for example in a thing that is away from the ground, the object will try to reach the ground, in this way minimizing the potential energy.

Now, if once the particle reaches the center of the square it remains at rest, it means that the total forces acting on the particle are zero and this is why the particle stays at rest, this would mean that the particle is in equilibrium, and if someone moves it a little bit of the center, some of the forces will increase and others will decrease, and then the equilibrium will be broken and the particle will move again.

In another case, if the particle is momentarily at rest (just for a few seconds) it may be because the forces acting on it are affecting the particle in such way that is moving is fully stopped in one direction, and the new forces are accelerating the particle in the opposite direction (in the same way that if you throw something upside when it reaches the maximum height it has for a brief moment a velocity equal to zero)