Boxberger attended Foothill High School in Santa Ana, California. He was drafted by the Kansas City Royals in the 20th round of the 2006 Major League Baseball draft, but did not sign, choosing to attend the University of Southern California (USC). For the USC Trojans baseball team, he was named a Second Team All-American by Rivals.com. Boxberger was a part of the USC rotation his freshman and junior year, and split his sophomore season between starting and relief. In 2007, he played collegiate summer baseball for the Orleans Cardinals of the Cape Cod Baseball League (CCBL), and returned to the CCBL the following summer to play for the Chatham A’s. He was named a league all-star in both seasons.

Boxberger was drafted by the Cincinnati Reds in the first round (43rd overall) of the 2009 Major League Baseball draft, and signed. Boxberger made his professional debut with the Lynchburg Hillcats of the Class A-Advanced Carolina League in 2010. He was named to the Carolina League All-Star Team. He was then promoted to the Carolina Mudcats of the Class AA Southern League. He had a combined win–loss record of 5–10 with a 4.91 earned run average (ERA) in 36 games, including 13 starts, as he was converted from a starter to a reliever after the promotion.

Boxberger began the 2011 season with the Mudcats, but was promoted to the Louisville Bats of the Class AAA International League. He finished the 2011 season with a 2.03 ERA and 11 saves in 55 games combined. He had the best strikeout ratio and lowest batting average against among all qualified Southern League relievers. Boxberger was the top Double-A closer in the minor leagues in 2011.

Boxberger, Yonder Alonso, Edinson Vólquez, and Yasmani Grandal were traded to the San Diego Padres for Mat Latos on December 17, 2011. Boxberger began the 2012 season with the Tucson Padres of the Class AAA Pacific Coast League. He was promoted to the Major Leagues on June 10, and made his debut the same day, pitching a hitless eighth inning against the Milwaukee Brewers in Miller Park.

Boxberger is considered to be a good fit for the late innings of games, either as a setup pitcher or closer. He has a four-seam fastball that ranges from 92 to 96 miles per hour (148 to 154 km/h), a changeup, a slider he throws at 88 miles per hour (142 km/h), and a 12-6 curveball he can throw from 78 to 81 miles per hour (126–130 km/h). Boxberger is tough on right-handers, posting a .225/.356/.354 line in 2012, but lefties hit .262/.380/.500 against him in the same span.

On January 22, 2014, the Padres traded Boxberger, Logan Forsythe, Matt Lollis, Matt Andriese, and Maxx Tissenbaum to the Tampa Bay Rays in exchange for Alex Torres and Jesse Hahn. During spring training, the Rays optioned Boxberger to the Durham Bulls of the International League. On May 8, 2014, Boxberger made an appearance in the top of the 6th inning vs the Baltimore Orioles with the bases loaded. He proceeded to strike out 3 batters on 9 pitches thus becoming the 2nd Rays pitcher ever to pitch an immaculate inning and the first pitcher ever on record to do so with the bases loaded. In 63 games, Boxberger finished with a 2.37 ERA with 104 strikeouts in 64.2 innings. He also saved 2 games.

In his first spring training appearance of 2016, Boxberger tore his adductor brevis muscle, requiring surgery. He began the season on the disabled list, returning to the Rays on May 30. In his first appearance of the 2016 season, Boxberger strained his left oblique muscle, and went back on the disabled list. He made 27 total appearances in 2016.

In the 2017 season, Boxberger was activated off the DL on June 20 after lat and forearm injuries. He was given a set-up role behind Álex Colomé. For the second straight season, he endured an injury plagued season, appearing in only 30 games.

On November 30, 2017, the Rays traded Boxberger to the Arizona Diamondbacks for Curtis Taylor. Boxberger won the closer competition in the 2018 spring training and served as the Diamondbacks closer. On September 10, he was removed as closer after a rough patch to begin the month. He had 32 saves at the time of his removal. He finished 3–7 with an ERA of 4.39 in 53 ⁄3 innings. On November 30, 2018, the Diamondbacks non-tendered Boxberger and he became a free agent.

On February 7, 2019, the Kansas City Royals signed Boxberger to a 1-year contract for $2.2 million, with $1 million in performance bonuses. Boxberger was designated for assignment on June 26 and released on July 3.

Boxberger signed a minor-league deal with the Washington Nationals on July 12, 2019. He was assigned to the Nationals’ Class-AA affiliate, the Harrisburg Senators. Boxberger was released on August 4, 2019, after pitching to a 1.04 ERA over eight games out of the Senators’ bullpen.

On August 5, 2019, it was announced that the Cincinnati Reds had signed Boxberger to a minor league deal and optioned him to the Triple-A Louisville Bats. Boxberger was released by the Reds organization on August 23, 2019.

On February 14, 2020, it was announced that the Miami Marlins had signed Boxberger to a minor league deal with a major league spring training invite.