Using the pen tool I can make an appropriate-looking dot. However, I then find that when I make another dot it tries to join them up, which is not what I want.
You use a different tool. You probably have the pen tool (tool for making paths) selected.
For an example, use the pencil tool instead or hold the left mouse button over the pencil tool in the menu to see what other drawing tools you have (see image below).
The pencil can be sized into a big square.
Alternatively, on a new layer, use the marquee tool, hold Shift (not sure of Mac alternative, I guess Command), and create a square. Then, use the gradient and fill it in. Then resize it as desired.