Jim, am I reading this right? That the brakes squeal when no pressure is applied to the lever? But do not squeal when actually braking?
I had an extremely loud squeal from the rear brake on my Fisher (Avid Juicy 5), and I tried many things to get rid of it, to no avail, until I tossed the stock Avid pads and put in Jagwire ones. Problem solved. And loud it was -- sounded like a 1950's city bus moving through the woods; animals and birds were fleeing in terror.
For Lynn's bike, there could be a number of causes -- contaminated pads or rotor; worn pads; bent rotor; calipers out of alignment; rotor bolts unevenly torqued.
Things to check: bent rotor (spin the wheel and look for a wobble); worn pads (there should be a couple of millimeters material left on each side). The other causes are more difficult to spot, so just read below....
Here are some things to try, with the caveat that I am not familiar with Shimano brakes (should be some cross-over, though):
1) align the calipers by loosening the mounting bolts, squeezing the brake lever 5 or 6 times, and then holding the lever (pressing the pads to the rotor) while torquing down the bolts to the specified value, alternating between bolts -- tighten one a little, then the other, back and forth until the torque wrench clicks (...you do own a small torque wrench, don't you? essential kit for any bike owner, esp. for carbon frames....)
2) clean the rotors with rubbing alcohol, methyl hydrate, or automotive brake cleaner; they may be contaminated with oil or grease; you can try to clean the pads too, using fine emery paper and alcohol wipe
3) loosen then re-torque the rotor bolts to their specified value, in the order required by the service manual
4) check the rotors for warp; if bent, try to straighten them (gently) using a pair of plumbers pliers; use a soft cloth to protect the rotor; or buy a rotor truing fork to do the job
5) replace the pads, if worn, or if you suspect they are contaminated but can't get them clean
Hope this helps, but post back here if none of this works. Good luck...
Svend