I think this picture best explains my issue:

First I translate the box
Creating a compound object whose centre will be the point about which the inner objects rotate is one obvious answer, and would be very quick to write. Just create an Object3D and add your box to it.
A similar approach is covered by this question. It shifts the point of the vertices for an object, so it effectively has a new centre.
Alternatively, you can mess around with the matrices by hand. Try this:
var boxGeometry = new THREE.CubeGeometry(boxEdge, boxEdge, boxEdge);
var mr = new THREE.Matrix4();
var mt = new THREE.Matrix4();
mt.setPosition(new THREE.Vector3(0,tubeRadius,0));
var box1 = new THREE.Mesh( boxGeometry, material );
box1.applyMatrix(mt);
var box2 = box1.clone();
mr.makeRotationZ(2 * Math.PI /3);
box2.applyMatrix(mr);
boxes.add(box2);
var box3 = box1.clone();
mr.makeRotationZ(4 * Math.PI /3);
box3.applyMatrix(mr);
boxes.add(box3);
boxes.add(box1);
scene.add( boxes );