Has 3.19.0-65 introduced new Secure Boot requirements to 14.04 LTS?
You are correct. The Canonical Kernel Team has enabled EFI_SECURE_BOOT_SIG_ENFORCE in the new 3.19 Ubuntu kernel.
That prevents from loading unsigned 3rd party kernel modules.
It looks like there is a script with GUI that is supposed to help to disable Secure Boot.
It did not work in your case. It depends on specific UEFI implementation in your computer.
But you can simply disable Secure Boot in your UEFI settings.
See this answer and comments below.
You can also disable Secure Boot running sudo update-secureboot-policy. This wiki page explains this method.