He played college football at Rice and was signed by the Houston Texans in 2014 as an undrafted free agent and has also spent time with the New York Giants.

[5][6] With 358 career points, he is second on the all-time Rice scoring list behind ex-NFL wide receiver Jarett Dillard.

[12] After participating in all four preseason games in which he missed multiple extra point attempts, Boswell was cut on August 29.

[19] Boswell made his NFL debut on October 12, completing all three of his extra points and making a 47-yard field goal attempt in a victory over the San Diego Chargers.

[21] He became the first player out of Rice to convert an extra point and field goal in a game since James Hamrick in Week 6 of the 1987 season.

[33] He also passed Kris Brown for the most made field goals by a rookie or first-year player in the history of the franchise with 29.

[36] On January 9, 2016, Boswell kicked four field goals in an 18–16 win in the AFC Wild Card Round game against the Bengals.

[39] His four field goals also tied for the most in Steelers playoff history (Gary Anderson in the 1989 Wild Card Round).

[40] On January 17, 2016, in a loss against the Broncos during the Divisional Round of the playoffs, Boswell tied a franchise record with seven total field goals completed in the post-season.

[52] Boswell made a game-winning 53-yard field goal in the next game to beat the Green Bay Packers 31–28 as time expired.

[54] The only longer successful attempt was in collegiate play when former Old Dominion kicker Jarod Brown made a 54-yard kick against the Pitt Panthers in 2013.

[57] Throughout the season, Boswell has kicked multiple game-winning field goals in the last minutes, earning him status as one of the Steelers' "Killer B's" and a spot on the AFC's Pro Bowl starting roster.

[64] During a Week 12 24–17 road loss to the Broncos, Boswell scored his first NFL touchdown on a fake field goal in the waning moments of the second quarter, tossing a two-yard pass to offensive tackle Alejandro Villanueva as time expired.

[68][69] Boswell was replaced by kicker Matt McCrane for the Steelers' final game against the Cincinnati Bengals for the 2018 season.

[79] On November 8, 2021, against the Chicago Bears, Boswell missed an extra point in the third quarter, but was 3 for 3 on field goals, hitting a 54-yarder, a 52-yarder, and a game-winning 40-yarder.

[82] During the season-opener at the Bengals, Boswell missed a 55-yard field goal that would have won the game in overtime which hit the left upright.

[86] Boswell made another postseason start when the Steelers traveled to Highmark Stadium to play against the Buffalo Bills.

During the game, Boswell successfully kicked two extra point attempts and made a 40 yard field goal as the Steelers ended their season losing 17-31.

[89] After an impressive start to the 2024 season, he was named AFC Special Teams Player of the Month for September.

[90] Boswell was named AFC Special Teams Player of the Month for October after converting 12 field goals.

[92] During the Steelers’ matchup against the Eagles, Boswell broke the team’s franchise record for most field goals in a single season.