When idling on your drive is the bike on centre or side stand?
Do you lose sparks to all plugs?
These intermittent probs are a bugger to sort most times because of the source of the problem can only be tracked whist it's playing up, if after 5 mins it fires up again does not give you much time to check/trace the problem.
Can't be a coil breaking down otherwise it would do it while idling/4K revs on driveway.
Does the bike have an immobiliser fitted?
Thinking back to a coil pack breakdown, I understand the FZS has two coil packs, one coil serves two plugs, would a FZS 600 actually run on just two cylinders/one coil?
It would appear by you experiments so far not to be heat related, the only other forces or elements involved are windforce, movement/rise and fall of suspension or flexing/swinging arm, steering.
Try sitting on the bike while on drive with the bike ticking over/4K revs, on the stand then off the stand then turning steering to the left and right, then bouncing up and down, if you do one of the actions at a time it should eliminated the possible issue. If one action does appear to be the fault try it several time to ensure it was not just coincidentle.
This is an interesting prob please keep us updated as the resolution will a good one for the FZS 600 armoury.