Egnew was born in Lubbock, Texas. He attended Plainview High School, and was a standout tight end for the Plainview Bulldogs high school football team. He was also a member of the Bulldogs basketball and track and field teams, and placed first in the state track championships in the long jump.
Egnew attended the University of Missouri, where he played for coach Gary Pinkel's Missouri Tigers football team from 2008 to 2011. He was a finalist for the John Mackey Award in 2010. Following his junior season in 2010, Egnew was named a first-team All-Big 12 selection and was recognized as a consensus first-team All-American. As a senior in 2011, Egnew's production fell, due largely to inconsistent play at the Tigers' quarterback position, but he was named a first-team All-Big 12 selection for the second consecutive year nonetheless.