Can the LG 768 be encypted? I have an LG 768 device. It is running Android 4.04 with Kernel version 3.0.21 and build number IMM76L. I can find no way to encrypt the device. None of the menus lead to an option to encrypt.

Has anyone figured out a way to encrypt this device?
Out of the box, No.

LG 768 User Manual

LG devices don't have the Full Device Encryption capability with the stock ROM, but you can always install an app to deal with that:

SSE - Universal Encryption




  Secret Space Encryptor (S.S.E.)  
  
  
  Password Manager, Message (Text) Encryption and File Encryption integrated in the all-in-one solution.  
  Store and manage all your passwords in one secure place protected with one Master Password.  
  Keep your messages, notes and other texts safe from unintended readers.  
  Securely encrypt your private and confidential files or whole folders.  Wiping (secure delete) feature is included.  
  Everything is encrypted using strong encryption algorithm: AES(Rijndael) 256bit, RC6 256bit, Serpent 256bit, Blowfish 256bit/448bit, Twofish 256bit and GOST 256bit ciphers are available.  
  Other Utils: Password Generator, Clipboard Cleaner, Algorithm Benchmark, ...  
  Minimal Permissions. No ads.  
  Cross-Platform (Windows, Linux, Mac OS X, ...) GUI Version and Command Line Version of the File Encryptor is available on the application home page (download section).  
  This software is Open Source project – we have nothing to hide, so you can have securely hidden everything you need.
  (as Open Source, this software is published under TSU - §740.13(e) encryption export license exception)