In general, a power switch can either be mechanically bypassed by shorting the power switch's cable pins inside the machine, or - more safely for your warranty - changing the power settings in the BIOS. I do not have information on the specific BIOS settings in the Euclid though.
One way to get automatic start-up is to set the BIOS to 'Wake On LAN' so that it can be activated remotely by a signal from a network. Another way is to enable a 'reboot after AC loss' type action, which is usually used to tell the PC to reboot when power comes back on after a power cut without having to press the power button. These options should be in the power management section of the BIOS settings.