Apple has made a slight -- but welcome -- change to how screen sharing works when a customer engages with an AppleCare representative. Previously, the rep could initiate a screen share if she or he thought it would help resolve the customer's issue more easily. The AppleCare representative would have the customer download a file, which would then enable them to share their screen.
Now, however, a customer can initiate screen sharing and the AppleCare representative must comply even if they don't believe sharing the screen will help resolve the customer's issue. The screen sharing tool is only available for live telephone calls and cannot be used while chatting with an AppleCare representative via the web.
Users who choose the phone option on support.apple.com can directly download the software needed for screen sharing right on the page:
Share your screen with Apple Support
You can download a small, temporary application and we will be able to see your screen during the support session. The download is automatic if we are calling you now, or follow the link in your confirmation email for scheduled support calls. If there is anything that you don't want Apple to see on your screen, remove it from view before your session begins. This session will be recorded for quality purposes. See the screen sharing terms & conditions for more information. Screen sharing is not required by Apple and you can end the session at any time.