GI Bill STEM Extension Act of 2014 - Authorizes the Secretary of Veterans Affairs (VA) to pay up to nine months of additional Post-9/11 Educational Assistance to a veteran who has used all of the Post-9/11 Educational Assistance to which he or she was entitled and who: (1) is enrolled in a postsecondary education program that requires more than the standard 128 semester (or 192 quarter) credit hours for completion in a specified science, technology, engineering, or mathematics (STEM) field; or (2) has earned a postsecondary degree in one of those fields and is enrolled in a teaching certification program.