Here's a physics-based way of looking at it.
The Coriolis force depends on the vector (cross) product of the angular velocity of the rotating frame and the velocity of the object. That means that it acts perpendicular to both of those vectors, and depends on the sine of the angle between them.
At the equator, there are two cases to consider. Moving north-south, the velocity is in the same direction as the angular velocity of the rotating frame (parallel to the earth's rotation axis). So the cross product is zero -- no Coriolis force.
Moving east-west, the velocity perpendicular to the angular velocity of the rotating frame, so you feel the full effect of Coriolis force. However, the force is perpendicular to the velocity and the axis of the earth -- in other words it acts in the local vertical, away from the centre of the earth. So it's not apparent in the usual way.
Strictly speaking, Eagle1's book should have said that at the equator there is no component of the Coriolis force in the horizontal plane.