How Do I Fix A Toilet That Is Making A High Pitched Sound After Every Flush?

The noise is what makes Plumber's Rich!
If it is symmetrical with the flow of water IT IS A LOSE WASHER THAT CONTROLS YOUR WATER FLOW! In the Water Closet behind your toilet, OR IN ONE OF YOUR faucets somewhere in the house! ANYONE OF THEM!
 
So start by replacing that tower that shuts your water off in the Water Closet! If it still is going on then go to the CLOSEST water faucets and tighten the washer on the bottom with a regular screw driver! COLD WATER SIDE ONLY! Till it stops! 100% positive!
And a PLUMBER told you that!
 
logic101

That happened to me and it sounds like you need a new pump. You can buy them at walmart or any store that has a plumbing section, they are not expensive and easy to put on. It's one of those situations that it's easier to replace then to fix the old one.

I would say maybe it is your tank just filling back up with water, if it stops right after.  If not you may need to check the pump or back of the toilet, or even check your septic system.  If worse comes to worse call a plumber for more advice, or ask someone who works at a store that sells plumbing supplies they may know more also.  So you don't have to hire a plumber.  Good luck.

Buy a new toilet or get someone to check it out it may cost you a fee pennies but it will worth your wild this is something you need to have done.