When I select Enclave from the START menu, my two monitors go black for a moment, and then they both revert back to my desktop. What do I need to do to get the game to run?

First of all you should provide more information about your system (hardware, OS, etc.) in order to get help.
Unfortunately the game is known for being unstable on modern systems.

That said, supposing you are using Windows 7 or later, you may try one or a combination of these possible solutions:
- Setting the game's executable to run in compatibility mode for Windows XP;
- Setting the game's executable to run using a single core (read here);
- Changing the default video rendering (read here).