Well I'm just making the basic shape. You can't select parts of it, because you can't split the box in two, so to speak, as you'd need two separate shapes.
I think what you're looking to do is an illustrator feature, where you divide the box into shapes, in this a diagonal line over the square, select both objects, using PATHFINDER to divide the shape.
I don't know what you're thinking with the Eyedropper tool? The eyedropper with the first click samples one objects properties and then the second click on another similar object pastes that objects style, whether it be a box or text or whatever object you need.
With the way I've described, you simply draw another object to mirror it and colour them any colour you want, removing the diagnol strokes of course with the selection tool. And the other suggestion there is a good idea too.
But I like the other idea. But I wouldn't move them. As they wouldn't be in line with each other. You can do a step and repeat, and don't put in a distance and copy once, the proceed with the point removals.
But interesting point, you can't do what you want to do easily.