It was decided that either the button was at fault, or electrical interference was causing the problem, but I can't be entirely sure which. The cable we used was shielded, but the earth wasn't connected at both ends, so the shielding wasn't doing anything - this will be fixed. And I'll replace the button itself anyway, just in case!
Martin also suggested using a resistor in the wiring, but I can't quite work out where he means - probably either because he's cleverer than me (and knows the technical jargon) or because of slight language differences over the pond. Anyway, this is what he said:
If interference is being picked up on the external trigger line, you could try adding an external resistor to increase the strength of the pull-up on the external start/stop input. Connecting a 1k resistor between the input and the 5v reference should do this.
Am I right in thinking that this will go inline in the wiring and not parallel to the switch (which would surely be enough to short the start/stop terminals).
Clarification hoped for
Tristan