This is probably why the click-to-copy share links don’t work with Flash 10:

Setting data on the system Clipboard requires user interaction

In Flash Player 9, ActionScript could set data on the system Clipboard at any time. With Flash Player 10, the System.setClipboard() method may be successfully called only through ActionScript that originates from user interaction. This includes actions such as clicking the mouse or using the keyboard. This user interaction requirement also applies to the new ActionScript 3.0 Clipboard.generalClipboard.setData() and Clipboard.generalClipboard.setDataHandler() methods.
What is impacted?

This change can potentially affect any SWF file that makes use of the System.setClipboard() method. This change affects SWF files of all versions played in Flash Player 10 and later. This change affects all non-application content in Adobe AIR?however, AIR application content itself is unaffected.
What do I need to do?

Any existing content that sets data on the system Clipboard using the System.setClipboard() method outside of an event triggered by user interaction will need to be updated. Setting the Clipboard will now have to be invoked through a button, keyboard shortcut, or some other event initiated by the user.

[From Adobe – Developer Center : Understanding the security changes in Flash Player 10]