The Form has two properties called MinimizeBox and MaximizeBox, set both of them to false.
To stop the form closing, handle the FormClosing event, and set e.Cancel = true; in there and after that, set WindowState = FormWindowState.Minimized;, to minimize the form.
Bind a handler to the FormClosing event, then set e.Cancel = true, and set the form this.WindowState = FormWindowState.Minimized.
If you want to ever actually close the form, make a class-wide boolean _close and, in your handler, set e.Cancel to !_close, so that whenever the user clicks the X on the window, it doesn't close, but you can still close it (without just killing it) with close = true; this.Close();
(And just to make my answer complete) set MaximizeBox and MinimizeBox form properties to False.
How to make form minimize when closing was already answered, but how to remove the minimize and maximize buttons wasn't. FormBorderStyle: FixedDialog MinimizeBox: false MaximizeBox: false