Based on that breaker and wiring configuration, it is a single-phase 240 volt connection. A three-phase breaker has 3 distinct sections.

The unconnected wire is (as you suspect) your neutral line. If a device operates as a single 240 volt device, neutral isn't used. If a device operates as "two" 120 volt devices, neutral is used.

So, if you have a power supply that supports a 240 volt connection, you can connect one hot to one input, the other hot to the other input, ground to ground, and you should be good to go!