Hi Mike,
It looks like QuickBooks doesn't like receiving an empty field after it has been populated. I did several tests based off what you said, and agree if you add a note in Method for a Customer, sync to QuickBooks, then try to blank out the note from Method, QuickBooks doesn't allow this blank value, so it keeps the original note. I did find however, if I remove this note in Method, but keep a single character, for example ".", then after a sync the note does get updated correctly in QuickBooks and Method. Now, if you were to delete the note completely from within Quickbooks it will sync over just fine, and both Method and QuickBooks will now have a blank note.
I am going to bring this up to our development team to see if there is anything Method can do about this, however I am not sure anything will be possible on our side. I will post here when I receive some feedback.
On a side note, you mention you added the notes field to your EditCustomer screen because it is not there by default. If you were to click the "More Actions..." button from the bottom of the stock EditCustomer screen, there is a "notes" option here that provides a pop-up screen with a large Notes area for that customer. I believe Method would have designed the screen this way because we didn't want the screen to seem cluttered with too many fields, and because this allows for a larger area to create the custom note.
-Ben