sounds like a grounding issue to me.
Microphonic feedback sounds like squeeling and whistling, and happens occasionally not constantly. Microphonic problems are solved by waxed potting but i dont think that'll help in this case.
if you say your strat (which also has single coils) DOESNT get the same noise in the same room, then it's not 60 cycle hum.
The jag has a complex wiring schematic - i think a wire has come loose and is not touching somewhere it should be. A grounding wire not properly hooked up the bridge would mean your guitar doesnt get grounded when you touch the strings - which it seems is your problem.
Does the sound happen equally on both and the lead and rhythm circuits?