When the water is turned on for toilet flush or sinks-hot or cold and when the clothes washer is filling (newer type washer that adds more water), many times for the first 4 minutes (starting the water and when the water add is stopped) there is a squeak in the plumbing. It seems to come from the utility room where the water enters the house (and where the house shut-off/pressure reducing valve is located). Again, even though the taps are turned on away from the utility room, the squeaky noise (high pitched) comes from the utility room.
It sounds like your problem is pipe movement.
During installation water pipe/tubing is suspended with metal tape, supported by brackets or run through holes drilled in the studs.
When the water flow abruptly starts or stops, this causes a pressure surge to run down the length of the pipe, causing it to move.
You will have to find the location where the squeak is coming from and modify the mounting so the friction causing the squeak is eliminated.
Rubber tape around the pipe under the bracket, repositioning, spray urethane foam can all be used to minimize the movement or separate the pipe from what it's rubbing against.
Installing water hammer arresters in the hot and cold runs can help absorb the pressure pulse and minimize the movement in a worst case scenario.
Our shower on the other end of the house was a good wash day indicator, creating a regular squeak symphony as the solenoid valves cycled to moderate water temperature. A couple squares of neoprene rubber and vinyl coated drive style brackets eliminated the issue by providing a semi-resilient attachment that separated the pipe from the sub-floor it was laying on.