The answer is yes. From the looks of the one-button clicker on the touchpad, it seems at first that a simulated middle-click is impossible. That is, clicking both the left and right buttons. However, you can indeed click both left and right clickers. Just click as far away from the middle as you can.
You’d have to be careful, though, not to press too hard, as you might break your clicker.
Middle-click is useful in Firefox for one-click closing of tabs or for opening links in new tabs (without having to press the Ctrl key). It’s also useful when copying and pasting across windows, especially when using the Linux terminal/console.
You can also middle-click by tapping two points on the touchpad. Yes, much like the MacBook’s trackpad, the Asus Eee’s touchpad supports multi-touch. So this means you can tap with two fingers, and that action will be read (by default) as a middle-click.