Lately, every time I use the washing machine or flush the toilet, a high pitched noise can be heard in the outside pipes for about 10 minutes. I can't hear the noise inside the house. What can it be? How can it be stopped?

While that high-pitched noise (HPN) is happening, open another faucet somewhere and see if the increased flow causes the HPN to go away or change in volume or pitch. If it does, then it is a flow issue at some narrow range of water flow somewhere in the supply plumbing. If it doesn't, then most likely is in the commode or wash machine valve. It may be more audible somewhere else, but it usually is in the valve itself. There often isn't much to do except replace the offending valve, but it is a lot cheaper to ignore it if you only hear it outside.

I know of a house that has a whine in one circulator zone. At least that one goes away in the summer. :^D

So no other fixtures being used can create this sound, right? Try all of your cold taps see if anything else can do it