You can try saving your symbology as a .lyr file, then importing it when you want to re-use the symbology:
After setting your preferred symbology (edit: you must use the 'Manual' Classification Method for this to work, as @michael-stimson notes), right-click on the layer in the Table of Contents and select "Save as Layer File." Note where this file is saved. The layer file contains a reference to where your data file is stored as well as the symbology that you have set.
You can use the layer file in two different ways:
- If you want to reference the same data layer and it's stored in the same place, just use 'Add Data" to add the .lyr file to your map document.
- When you have any layer that you want to use the same symbology settings, you can go to the Symbology tab on the Layer Properties and use the "Import" button (looks like a folder with arrow -- see image below), then select your .lyr file that you just saved. This should import your manually-set class breaks and colors.