This is basically a partially-automated version of Naldi Stefano's tutorials here, here, here and here, which were partly based on blog posts by Pavel Kogan here and here. All credit goes to them for figuring out how to do it; I just made it into a script.
The Linux Mint 18 or 18.1 installer has an option for installing on
LVM inside an encrypted LUKS container, but this is only offered if
you want to erase the whole disk (no dual boot), and also leaves the
/boot partition unencrypted. If you want to encrypt everything
including /boot, or want to install any sort of encrypted system
alongside another OS for dual boot [1], you have to configure the
bootloader and initramfs manually -- which is time-consuming and easy
to get wrong. This script guides you through the process, and
automates as many of the commands as possible, making it much easier
to set up.
This is still an advanced configuration, though, and assumes you are comfortable with the terminal, shell scripts, partitioning, LVM, and installing and managing normal non-encrypted Linux systems. If in doubt, read the tutorials linked above, and above all, PLEASE BE CAREFUL. Typing anything wrong here could erase your hard drive! Make sure you test anything in a virtual machine before trying it on your real machine.
[1] You can install multiple Linux systems inside the same encrypted
container (using different logical volumes as /), and the bootloader
will pick those up just fine. But if you want to dual-boot alongside
something else (e.g. Windows), that has to use an unencrypted
partition outside the container.
Boot the Mint live USB as normal. But instead of running the 'Install Linux Mint' launcher on the desktop, open a terminal and run the following commands:
sudo apt-get -y install git
git clone https://github.com/callumcameron/mint-encrypted-install
cd mint-encrypted-install
./mint-encrypted-install
This will guide you through the rest of the process.