If you are on this site, then you have access to the video tutorials. I would look at the downloading and setup, and the ports and pins ones, this will give you a lot of information as to how the system runs from a PC. It may be that you can figure out how your "pupil" has managed it.
I assume the Bridgeport was manual, and stepper motors have been added, and also some facility to start and stop the spindle motor. If the Bridgeport was CNC previously and all that has happened is that you have replaced the Bridgeport system with a PC running Mach, then it could be different - BUT
The standard method of running Mach3 is via the printer port cable (the 25 pin D output). The printer cable is run from three addresses inside the PC and these three adress run three sets of pins. 2 - 9 is a block of 8 output pins used for driving the stepper motors. pins 10 - 13 and 15 are a set of input pins from sensors on the machine. pins 1,14,16 and 17 are another set of output pins and these are usually used to control relays to run the spindle, coolant etc. pins 18 to 25 are signal ground connections.
It is likely he will have used some form of breakout board from the cable - (either a commercial one or homemade) and from there you should be able to identify the pin numbers. If you look on the configure/ports and pins menu on Mach3 it will tell you how it has all been set up. The answer to your problem may be that when installing the new software - which, I assume may be Mach3 - the pin that controls the relay to start the spindle is no longer assigned properly on the configure file.
You say the problems were intermittent - this could be due to all sorts, but the main thing is having a reliable signal return on your printer cable (there are 7 so you have enough to us one each from every application)
I don't know what else to say until I know more of the problem - so have a look at your set up and see if my ramblings get you any further.
To be fair to Art - Mach3 is extremely reliable, so if it is connected up properly - it tends to work very well.