If your computer has a Windows 8 sticker on it, you need to turn off Secure Boot so your BIOS will recognize any OS beyond Microsoft's preference.
Unless you have corrupt media (you can also reboot and select on the disks menu Mint's CD integrity checker), it sounds like that's the issue. Mint probably has keys you get around this, but your better off turning of Secure Boot. I found some Asus documentation on how to do that: http://acer--uk.custhelp.com/app/ans...le-secure-boot
There also may be something going wrong with EFI configuration set up on it. Turning off all of that unless you have funky hardware is probably your path of least resistance.
For a background on what Secure Boot is, Microsoft made part of their Windows sticker certification that all new 8 PCs have a "feature" called Secure Boot that checks keys of installed operating systems. It's *cough, cough* intent to keep malware that may embed its in the boot process from starting up. A side effect (cough, benefit, getting over a monopoly cold) of it is that it prevents the user from reverting to an older version of Windows or an alternative OS without a proper key that the OS distributor has to pay for (SOL with secure boot and custom compiled kernels). Microsoft kindly took away the requirement for manufacturers to include an option to turn Secure Boot off with the Windows 10 sticker certification.