If you have a version of OSX with Apple gatekeeper, you may need to follow these steps to use ZC. THis is an Apple issue, not a zC issue.

Back up any quests, and .sav files.
Delete any folder that you tried to set up, that contains Zelda Classic and ZQuess; such as /Applications/Zelda Classic.
Download the current version of Zelda Classic. At the time of this post, it is 2.50.2--not 2.50.3.
Extract the archive file in your Downloads folder.
Manually create the Zelda Classic folder in /Applications again. (If you do not have access to the global Applications folder, your local applications folder should still work for this purpose.)
Drag the individual Zelda Classic and ZQuest apps into your newly created folder. DO NOT move the folder containing the apps over!

^ Based on the steps provided by @Wolfman2000