Mincey was born in Statesboro, Georgia in 1983. He attended Statesboro High School, where he was a letterman in basketball and football for the Statesboro Blue Devils, but only began playing football as a junior. Mincey graduated from Statesboro High in 2002.
After accepting a scholarship from Jacksonville State University, he decided to attend Butler County Community College in El Dorado, Kansas, where he played from 2002 to 2003. He was recognized as a first-team all-conference, after tallying fifty-nine total tackles, 11.5 tackles for loss, four quarterback sacks, one interception, two pass breakups, three forced fumbles, four recovered fumbles and one sack for a safety. Rivals.com rated Mincey as the No. 18 junior college player in the country.
After his sophomore year, he accepted an athletic scholarship to transfer to the University of Florida in Gainesville, Florida, where he played for coach Ron Zook and coach Urban Meyer's Florida Gators football teams in 2004 and 2005. As a senior team captain in 2005, Mincey earned second-team All-Southeastern Conference (SEC) honors when he ranked fourth on the team with sixty-two tackles and second with 10.5 tackles for a loss. He started in all twenty-four games in which he played for the Gators.
Mincey graduated from the University of Florida with a bachelor's degree in Health and Human Performance in 2007.
The New England Patriots selected Mincey in the sixth round (191st pick overall) of the 2006 NFL Draft,. He was used mainly as a linebacker before being released on August 31.
He was signed to the San Francisco 49ers on September 3, 2006.
Mincey was signed by the Jacksonville Jaguars off the San Francisco 49ers' practice squad on December 13, 2006, to play as a defensive end. After suffering from injuries during the preseason, he was waived on August 26, 2007 and later signed to the team's practice squad. On November 8, 2007, he was signed to the active roster and got a chance to play in six regular season games and two playoff games, finishing with twelve tackles (seven solo), one quarterback sack and six quarterback pressures.
In 2008, he started the season on the physically unable to perform list with a wrist injury and saw action in only three games after being activated on November 24. Mincey was waived injured on August 26, 2009 and was out of football during that season. He was re-signed for the 2010 season, eventually starting at left defensive end over former first-round draft choice Derrick Harvey and playing in fifteen games (8 starts), while compiling five sacks (led the team), twenty five quarterback pressures (second on the team), thirty-one tackles and three pass deflections.
He started all sixteen games and compiled eight sacks during the 2011 season, which led the team. Faced with free agency, Mincey signed a new four-year contract with the Jaguars on March 13, 2012. He was released on December 12, 2013.
Mincey was signed as a free agent by the Denver Broncos on December 16, 2013, reuniting him with former Jacksonville coach Jack Del Rio, who was the team's defensive coordinator. He was used as a reserve player and was a part of the playoff push to reach Super Bowl XLVIII.
On March 11, 2014, he signed with the Dallas Cowboys as a free agent, who were looking to improve their defensive depth after being forced to release DeMarcus Ware for salary cap reasons. He quickly developed into one of the Cowboys leaders and was named a team captain after Justin Durant was lost for the season. He led the team with 6 sacks and 39 quarterback pressures, after starting every game and keeping second round draft choice Demarcus Lawrence in a reserve role.
In 2015, Mincey missed the first four days of training camp in a contract dispute. He started six of the first 13 games, including four that Greg Hardy missed while he was suspended by the league. His playing time and effectiveness decreased during the year, after he was limited by bone spurs in his right elbow, that eventually required offseason surgery. He didn't register a sack during the season and was declared inactive against the Buffalo Bills. Besides playing defensive end, he was also used at the one-technique defensive tackle position in passing situations.
Mincey announced his retirement from professional football on August 23, 2016.
In 2012, Mincey opened his own recording studio in Jacksonville, Fl. named, "Mr. Mince Recording Studios". . He is currently managed by OG Ron C.