For those who can't see the images again, here are the links from the page source:
http://www.fileden.com/files/2008/10/14/2142989/improv.tiff
and:
http://www.fileden.com/files/2008/10/14/2142989/nasty.tiff

Might I suggest using a format more common on the web, like jpg?

In regards to your question, I am not sure what direction you are looking at the cube from in the second picture.  Are you looking directly toward a corner or toward a face?  If you are looking toward a face and getting this effect, then it is probably because the face is made up of four triangles with a central vertex.  Since shading on polygons start from the vertices, and since the light source is over that central vertex, you get the effect in your second image.

If that is the case, I would reduce the number of triangles you are using on that face.  There is no reason for a face to have four triangles instead of two.  I'm assuming you created this cube using jPCT's Primitives.getCube method, which has four triangles on at least two of the sides for some reason.  You could try building your cube manually, only using two polys per side.  Something like this should work (untested, but you get the general idea):
    // Creates a cube of the specified color
    private Object3D coloredCube( float scale, Color color )
    {
        TextureManager.getInstance().addTexture( "CubeClr",
                                               new Texture( 4, 4, color) );

        // 'offset' is simply half the length of the cube's side:
        float offset = 0.5f * scale;

        // Create a new object with the proper number of polys:
        Object3D obj = new Object3D( 12 );
        // Define the object's polys:
        obj.addTriangle( new SimpleVector( -offset, -offset, -offset ), 0, 0,
                         new SimpleVector( -offset, offset, -offset ), 0, 1,
                         new SimpleVector( offset, offset, -offset ), 1, 1,
                         TextureManager.getInstance().getTextureID( "CubeClr" ) );
        obj.addTriangle( new SimpleVector( offset, offset, -offset ), 1, 1,
                         new SimpleVector( offset, -offset, -offset ), 1, 0,
                         new SimpleVector( -offset, -offset, -offset ), 0, 0,
                         TextureManager.getInstance().getTextureID( "CubeClr" ) );
        obj.addTriangle( new SimpleVector( offset, -offset, offset ), 0, 0,
                         new SimpleVector( offset, offset, offset ), 0, 1,
                         new SimpleVector( -offset, offset, offset ), 1, 1,
                         TextureManager.getInstance().getTextureID( "CubeClr" ) );
        obj.addTriangle( new SimpleVector( -offset, offset, offset ), 1, 1,
                         new SimpleVector( -offset, -offset, offset ), 1, 0,
                         new SimpleVector( offset, -offset, offset ), 0, 0,
                         TextureManager.getInstance().getTextureID( "CubeClr" ) );
        obj.addTriangle( new SimpleVector( -offset, -offset, offset ), 0, 0,
                         new SimpleVector( -offset, offset, offset ), 0, 1,
                         new SimpleVector( -offset, offset, -offset ), 1, 1,
                         TextureManager.getInstance().getTextureID( "CubeClr" ) );
        obj.addTriangle( new SimpleVector( -offset, offset, -offset ), 1, 1,
                         new SimpleVector( -offset, -offset, -offset ), 1, 0,
                         new SimpleVector( -offset, -offset, offset ), 0, 0,
                         TextureManager.getInstance().getTextureID( "CubeClr" ) );
        obj.addTriangle( new SimpleVector( offset, -offset, -offset ), 0, 0,
                         new SimpleVector( offset, offset, -offset ), 0, 1,
                         new SimpleVector( offset, offset, offset ), 1, 1,
                         TextureManager.getInstance().getTextureID( "CubeClr" ) );
        obj.addTriangle( new SimpleVector( offset, offset, offset ), 1, 1,
                         new SimpleVector( offset, -offset, offset ),1, 0,
                         new SimpleVector( offset, -offset, -offset ), 0, 0,
                         TextureManager.getInstance().getTextureID( "CubeClr" ) );
        obj.addTriangle( new SimpleVector( -offset, -offset, offset ), 0, 0,
                         new SimpleVector( -offset, -offset, -offset ), 0, 1,
                         new SimpleVector( offset, -offset, -offset ), 1, 1,
                         TextureManager.getInstance().getTextureID( "CubeClr" ) );
        obj.addTriangle( new SimpleVector( offset, -offset, -offset ), 1, 1,
                         new SimpleVector( offset, -offset, offset ), 1, 0,
                         new SimpleVector( -offset, -offset, offset ), 0, 0,
                         TextureManager.getInstance().getTextureID( "CubeClr" ) );
        obj.addTriangle( new SimpleVector( -offset, offset, -offset ), 0, 0,
                         new SimpleVector( -offset, offset, offset ), 0, 1,
                         new SimpleVector( offset, offset, offset ), 1, 1,
                         TextureManager.getInstance().getTextureID( "CubeClr" ) );
        obj.addTriangle( new SimpleVector( offset, offset, offset ), 1, 1,
                         new SimpleVector( offset, offset, -offset ), 1, 0,
                         new SimpleVector( -offset, offset, -offset ), 0, 0,
                         TextureManager.getInstance().getTextureID( "CubeClr" ) );
        obj.build();
        return obj;
    }

A code sample would be helpful, but my first impression is that you have the ambient light up too high.  Try adjusting the brightness of both the ambient light and the point light source.

On a separate note, for some reason, your pictures didn't show up in my browser (Firefox 3.6.10, Ubuntu Lucid 64-bit).  I had to grab the links from the page source.  I'm guessing it is due to the fact that they are in .tiff format.  In case anyone else has the same issue, the links are:
http://www.fileden.com/files/2008/10/14/2142989/no_ref.tiff
and:
http://www.fileden.com/files/2008/10/14/2142989/reflecting.tiff

Ok, give this a try (to see if my logic about flipping x and z in camera space to rotate the scene properly is incorrect).  Set up your scene as you did earlier where you can see the characters from looking sideways at the scene (i.e. comment out matrix thing, FOV to 45, position of camera -y/z flipped and camera.lookAt to the position of Max's camera's target).  Then call the following method:

    public void rotateScene90( Object3D[] sceneObjects, SimpleVector lookTarget, Camera camera )
    {
        SimpleVector sourcePosition, targetPosition, translation;
        for( int x = 0; x < sceneObjects.length; x++ )
        {
            // convert object's position into camera space:
            sourcePosition = new SimpleVector( sceneObjects[x].getTransformedCenter() );
            sourcePosition = sourcePosition.calcSub( camera.getPosition() );
            sourcePosition.matMul( camera.getBack() );
            // determine the new position (in camera space):
            targetPosition = new SimpleVector(sourcePosition.z, sourcePosition.y, sourcePosition.x );
            // convert new position into world space:
            targetPosition.matMul( camera.getBack().invert3x3() );
            targetPosition.add( camera.getPosition() );
            // translate to the new position:
            sceneObjects[x].translate( targetPosition.calcSub( sceneObjects[x].getTransformedCenter() ) );
        }

        // convert the look-target into camera space:
        sourcePosition = new SimpleVector( lookTarget );
        sourcePosition = sourcePosition.calcSub( camera.getPosition() );
        sourcePosition.matMul( camera.getBack() );
        // determine the new look-target (in camera space):
        targetPosition = new SimpleVector(sourcePosition.z, sourcePosition.y, sourcePosition.x );
        // convert the new look-target into world space:
        targetPosition.matMul( camera.getBack().invert3x3() );
        targetPosition.add( camera.getPosition() );

        // look toward the new look-target:
        SimpleVector look = targetPosition.calcSub( camera.getPosition() ).normalize();
        SimpleVector up = camera.getYAxis();
        up.scalarMul( -1 );  // Bug fix, shouldn't be necessary?  Try removing if scene is upside-down
        camera.setOrientation( look, up );
    }
Notice the bug fix for the setOrientation issue.  Depending on the scene, the call to up.scalarMul( -1 ) may or may not be necessary (remove it if the scene winds up upside-down).

That is very strange.  I can't imagine a scenario where you can see something then flipping the x and z coordinates in camera space of that thing plus the x and z values of the camera's look direction in camera space and no longer being able to see the thing - seems like that should always result in the equivalent of rotating the thing 90 degrees around its y-axis.  Obviously such a scenario exists in your case, so I guess it is just a limitation of my imagination..

Well, jPCT's up axis is -y, whereas worldspace up for max is +z. I accounted for that. Or do you mean something else?

No, I mean more like the x-axis you are using is equivalent to jPCT's z-axis.

Currently, there is no method for creating an empty/black texture of a specified size with alpha, something like this:

myTexture = new Texture( width, height, true );
This would be useful for cases where a texture's contents are generated entirely through an ITextureEffect implementation where alpha is used (i.e. if there is no "initial" frame to use when instantiating the Texture).  This is not high-priority for me, though.  I came up with a workaround that seems to function the same way:

BufferedImage image = new BufferedImage( width, height, BufferedImage.TYPE_4BYTE_ABGR );
myTexture = new Texture( image, true );

One billboard quad could be the child of a second billboard quad, both with the same center/pivot.  The polys of the outer layer would consist of vertices with lower values for z (or higher, depending on which side billboarding points toward the camera in jPCT) than those of the inner layer.  Then you would give the parent object normal transparency and the child object additive transparency.

--EDIT--
I should point out in case you haven't noticed already, jPCT automatically calculates the center/pivot of a quad to have the same z-value as the polys when the object is built.  One work-around to "trick" jPCT into using a different center is to add two additional back-faced polys (on the other side of the imaginary "cube").  In my above scenario you could leave the parent object as 2 polys and just add a the extra polys to the child object.  An alternative to adding polys is with translateMesh, I believe, however as I recall it will just revert back if you call build or buildAll afterwards.