Yes, you can use "expand". But that's a broad term. The specific term for this action is maximize. This is when the window "expands" to cover the entire screen.
Two related terms are minimize and restore. When you shrink and hide the window by clicking the
_ button, you minimize the window. When it's hidden and you bring it back, that's called restoring the window.
By the way, I call them windows, not screens. I'm not sure if "screens" is acceptable. Also, the "on" is not necessary, and, as @amI points out, the double squares "restore down" (also "restore" or "unmaximize") the window. The single square maximizes the window. So if you want to maximize the window, you could say
Please, maximize the window by clicking the single square
Anyway, these are the terms I know as a Windows user. You can see their usage in these pages: