Question

 Can three particles be in equilibrium under the action of their mutual gravitational force? Can one of the three particles be ?

Answer 1

Since gravitational force is always attractive. Three particles can never be in equilibrium, but one of them can..

Answer 2

1. possibility
If the particles are touching each other and they are arranged like a closely packed triangle, then they can be in a static equilibrium. 
====================== 
2.  possibility
Since the force is only attractive, and there is no other force, that is only situation of equilibrium it seems.   If they are free to move, due to attraction, they accelerate towards each other, so their speeds increase.  In case they are to be moving and they should be in equilibrium, then they should be at the same distances of each other.  It means they must move in a circular orbits.  Then they have centripetal force acting on them. 

If one particle is at the center of a circular orbit in which the other two particles are revolving and are always at the opposite ends, then the situation continues.  They can move with fixed speeds.

      See diagram.  Let there be three equal masses.  Let the middle particle be at rest. Due to the gravitational forces from other two particles cancelling, the will stay at rest.  For the particles in orbit moving with a uniform speed v, 
 
Centripetal force is supplied by the gravitational force due to other two planets.

       m v² /R = Gm²/R² + Gm²/4R²
            v = √(5Gm/4r)

============================
3.   possibility  -  static
If three of them are in a straight line and are static, then the middle particle can be in a static equilibrium.  But the two particles at the extremes need to be balanced by external forces to keep them at rest.