Startup button on front panel?

You are talking about a desktop, not a laptop... Laptops don't have a front pannel.

Those springs are probably only there for the cover of the power button, not for the switch itself.

As for the wiring of that switch, it's a simple, single contact, non-polarised switch. This mean that the two wires are equivalent and can be switched without any problem.

To find where it connect to the motherboard, you need to consult the motherboard documentation.
The motherboard's brand and model are normaly printed in white on the top face of the motherboard. You only need to remove the left side pannel to be able to see this.
With that information, navigate to the web site of the maker of the motherboard. Go to the support or downloads section. Download the documentation for your board that is available as a free download in the PDF format.

Electro
March 2014