I don't know about the ColorMunki but it was pretty simple with my old Spyder --
Since I believe the OP asked about two separate computers (one belonging to a friend) then sure, just calibrate using a separate process (don' try to share the ICC profile, which is monitor-specific.
On one machine with dual monitors, as has been said it depends on the graphics hardware. On the desktop PC (Windows) I had a card with multiple ports with separate LUTs and the process was simple. In Windows all you need to do is use the tool to swith the displays, and in the calibration software you can switch to calibrating another display, NO NEED TO UNPLUG A MONITOR!
Then when in Windows you switch displays the software will switch to the appropriate ICC profile.
Like I said, simple. Things get a bit complicated if you are calibrating multiple monitors off a single LUC, because you have to have a separate ICC profile for each monitor but to switch you have to manual load the needed profile. This is common with laptops because they often only have a single LUC, even though they often have a port for an external monitor!