Our 2004 Honda Odyssey is making a squealing sound in reverse (with or without foot on brake… but not every time for either). For a week, it made the squealing sound going forward, but it has now stopped that. We did recently have our tires rotated at Jiffy Lube… could a bad rotation job be making the squeal. Or is it something I can safely ignore 
New sounds or symptoms should never be ignored.
I assume that you have already checked all fluids, following the emergence of this problem, and I assume that you have found them to be normal.
I strongly suggest that you take the car to a REAL mechanic so that he can see what damage might have been done by the kids at Jiffy Lube.
With the hood open, and your other half’s foot on the brake, put it in gear, and see if you can hear the sound. He/She may have to tap the gas slightly (not enough to lurch forward and run you over - that’s a marriage counselor’s job). You can try this with it in park, too. Hopefully you can hear/find it.
To me, it sounds like this may be just a squealing belt, and a replacement belt/tensioner could fix it.
It’s pretty unlikely a mere tire rotation would cause this…unless you’ve got a bad rim on there. I’ve seen miss-matched rims rub against calipers, etc.
You didn’t tell us mileage, or when the last time the transmission was serviced. This is most likely not connected to the tire rotation, though.
Chase
You might save a great deal of money and avoid a great deal of hassle if you get to a reputable shop sooner rather than later for a check up. My guess is that the brake wear indicators are trying to tell you it’s time for new pads.