Prior to playing professionally, Enright attended St. Mary's High School and then Pepperdine University. In his first year at Pepperdine, 2005, he went 10-1 with a 4.62 ERA in 18 starts. The following year, he went 13-2 with a 4.05 ERA in 21 games. After the 2006 season, he played collegiate summer baseball with the Brewster Whitecaps of the Cape Cod Baseball League. In 2007, he went 12-5 with a 1.99 ERA in 18 starts. He was drafted in the second round of the 2007 amateur draft by the Arizona Diamondbacks.
Professional career
Arizona Diamondbacks
Enright began his professional career in 2007, splitting the season between three teams - the Yakima Bears, the South Bend Silver Hawks and the Visalia Oaks. He did not allow a single earned run in 15 combined innings of work, and he struck out 17 batters. In 2008, he pitched for the Oaks, going 12-8 with a 4.44 ERA in 29 starts. With the Mobile Bay Bears in 2009, Enright went 10-9 with a 3.98 ERA in 27 starts. He began 2010 with the Mobile Bay Bears, going 4-1 with a 2.88 ERA in 14 starts prior to his call-up to the major leagues. In 2011 while he was pitching for the Reno Aces, Enright threw a pitch behind Josh Donaldson of the Sacramento River Cats and was promptly ejected along with Donaldson as the benches and bullpens cleared.
On July 24, 2012, Enright was traded to the Los Angeles Angels of Anaheim for cash considerations. On Tuesday, April 30, 2013 Enright surrendered a two-run walk-off home run to Oakland Athletics slugger Brandon Moss in the nineteenth inning. The home run, which ended the longest game in Angels history, took place over six and a half hours after the first pitch, which was thrown by the Athletics' Dan Straily 7:05PM PT the previous day. He was designated for assignment on May 16, 2013. He was outrighted to AAA on May 19, 2013. He elected free agency on November 4, 2013.
Enright signed a minor league deal with the Philadelphia Phillies on January 24, 2014. He was released in July after posting a 5.58 ERA in 19 starts for the Lehigh Valley IronPigs. He had been 1-7 with a 7.57 ERA in his last nine games started with the IronPigs.
Enright signed a minor league deal with the Los Angeles Dodgers on July 28, 2014. He appeared in 7 games for the AAA Albuquerque Isotopes, where he was 0-4 with an 8.62 ERA. The Dodgers released him on April 5, 2015.
On July 16, 2018, Enright signed a minor league deal with the Arizona Diamondbacks. He elected free agency on November 2, 2018.
Coaching Career
Enright announced his retirement Jan. 9, 2019, and will now work as the pitching coach for the Hillsboro Hops, a minor-league affiliate of the Arizona Diamondbacks in the Northwest League.