One Button To Open Taskbar Shortcuts

I was using a mouse that had some extra buttons on it and one of the buttons was set so that when I clicked on a taskbar shortcut, the taskbar shortcut opened a new instance of the program even when there are already windows of that program opened. Say you already have a few Chrome windows opened. Normally, when you click on that program, if first wants you to select whether you would like to open one of the existing program windows, a new window, unpin the program from the taskbar, etc. But with this button it will just take you into a new program window.

I found this very convenient. Can this feature be set up on most 3+ button mice?

Is there some way to use it with a 2 button mouse, too? My keyboard has a trackball built in but there are only 2 buttons. I'd be fine with changing one of the keyboard buttons to this. Or maybe holding down the function key and clicking one of the mouse buttons.

If you are curious, the mouse that I was using:
Prestige - Evoluent VerticalMouse 4 Wireless Laser Mouse
I bought it at best buy.

 

That seems to be more for configuring the extra buttons on your mouse, whereas I am wanting to mimic the mice that have extra buttons.

Sounds like there is generic software for input devices to configure these kinds of things. So, I will probably just have to hunt around a bit later.