My users go to set the date for a task and the column appears on the left side with cached dates, so if they want to set the reminder for Monday or Tuesday, it blocks off the days and they either have to do it manually or click and change.
Is this just the Salesforce Autocomplete for the Global Search and Lookup fields that is turned on or off for every user, or can I control is another way?
To permanent remove this, you can change the browser settings. From the screenshot, I cannot guess which browser you are using but if you using chrome, remove existing cache date by going to Chrome Settings >> Privacy & Security >> Clear Browsing Data >> Auto fill form data. After this, go to auto fill settings and switch off the auto fill forms setting.