What does the X symbol mean at wire junctions?
In this case, the X is there to let you know that there is no wire connection. If you study the schematic in your link, you will see that each place where there is an "X" one of the lines is NOT a wire. It is a part of a box or dividing line that has been put in place for organizational or documentation purposes.

I would rather see it be a dashed or dotted line, personally. Or in this day and age, simply have the non-wire lines be a different color from the wire lines. But this is probably an old schematic.

I think the white rectangle you are talking about is the place where you plug in the power supply. I think it is trying to show you that the power supply socket has a built-in switch, and that the internal batteries are disconnected when you plug in an external power supply. This prevents you from trying to charge the internal batteries when you plug in the supply. The logic is not clear from the diagram (looks almost opposite) but that would make no sense. It has to disconnect upon insertion, logically.