Secure boot is a feature of UEFI replacement for BIOS and can be turned off. Mint doesn't appear to work with secure boot, but I believe there are Linux distros that do. You'll need a 64 bit distro to work with UEFI.
FWIW I've had a UX21E Zenbook for two years now and love it. The keyboard is very ordinary, but the display and battery life are great. Purchased mine as a refurb for 40% less than retail.