Should I use the Color navigator software that came with the Eizo monitor...
yes! to keep it simple: Color Navigator "knows" the monitor... the software contains certain correction tables for that particular monitor other software do not contain.
should I install the Color navigator software first, then the Colormunki software, or does it even matter what order they are installed in?
either way. Actually you don't have to install the Colormunki software at all (unless you need it for printer profiling or so...).