Have any signal processors in your signal chain (especially compressors)? Do you run with a lot of high frequency boost or bright switch on? Using a wireless system?
One cause of problems like this can be when the frequency response capability of your gear extends beyond the frequencies you're really using. This is especially a problem with compressors. In addition to using the intentional sound of your bass to generate a control signal for the compressor, it may inadvertently hear high frequency noise that's outside the useful bass frequency range (and sometimes beyond human hearing range). Even though you don't hear this high frequency hash, the compressor sees it and pumps the gain up, which affects the audible part of your tone. If you play with a lot of treble boost (I don't expect a lot of Spyders have 3-year old flatrounds and an Ampeg B-15!), then when the gain is bumped up by something you can't hear, it can cause feedback in the stuff you can hear. In your case, it's probably not physical feedback that you're experiencing, it sounds like the pickup coil might be oscillating.
Wireless units almost always have a compressor, and the often seem to have this problem of overly extended frequency response.
When you bump the stereo switch, you interrupt the signal enough for things to settle down again, although it's just a matter of hitting the right note or volume level to trigger the feedback again.
There's a bunch of things to try. First, does the problem go away if you cut your treble significantly? Next, I'd try bypassing effects in your chain, one by one to see if the problem is due to a single unit (if it is, it will almost certainly be a compressor or distortion unit). Does the squeak go away if you tap the pickup?
If any of these things have an effect, the first thing you probably want to do is rig up a lowpass filter that gets rid of the high frequency junk. Your playing isn't generating any critical signal at 10KHz anyway, so lopping stuff above that point out (or even at 5KHz) won't really have much effect on your sound. But it will kill the junk that's confusing your compressor. Obviously changing compressors may improve the problem as well.
If tapping the pickup stops the squeak, then you might see if Alembic will replace the pickup - it might be slightly microphonic. But even if they would do that, you really probably will be better of blanking out the parts of your signal that aren't meaningful.
I don't know exactly what's in the Series II high frequency noise reduction mod (not applicable to your bass), but I wouldn't be suprised if this is part of it.
Good luck
David Fung