FROM THE SYNERGY FAQ:

How do I disable or modify Synergy's automatic version checking feature?

By default, Synergy automatically checks once per day to see if a new version has been released. Your privacy is protected because no information is transmitted to the server in order to perform this check; Synergy merely asks the server what the latest version number is, and if the number is higher than the currently running version, a dialog is displayed.

There is not yet a control in the user interface for this feature, but you can still control it by editing the Synergy preferences file. Make sure Synergy is not running when you edit the file. The file is at "-/Library/Preferences/org.wincent.Synergy.plist". To modify the settings look for the following entry in the file:

And etc.

NEXT!