Henry has played for Canterbury in New Zealand domestic cricket since 2011, making his first-class cricket debut in the 2010–11 Plunket Shield against Wellington in March 2011. He has played county cricket in England for Worcestershire for a time in 2016[4] and for Derbyshire in the 2017 NatWest t20 Blast[5] and played for Kent as their overseas player in the first half of the 2018 season before agreeing to return to play at the end of the season for the club.[4][6] After taking seven wickets on his Kent debut against Gloucestershire, Henry took his best innings and match bowling figures against Durham in late April 2018. He took five wickets in Durham's first innings and seven in their second to record his first ten-wicket haul with match figures of 12/73.[7][8] Henry was awarded his Kent cap during his first stint with the club. In November 2021, Henry was again signed to play for Kent, this time for the 2022 cricket season in England.[9]
In July 2019, he was selected to play for the Edinburgh Rocks in the inaugural edition of the Euro T20 Slam cricket tournament.[11][12] However, the following month the tournament was cancelled.[13]
In February 2023, Henry was signed by Somerset for the County Championship until the following July.[14] Henry had a very successful season with Somerset by being the tournament leading wicket taker and taking the winning wicket in the final against Essex to secure Somerset their first T20 title since 2005.[15]
Despite not being picked in the final squad of 15 for the 2015 Cricket World Cup, he was selected as a replacement for Adam Milne in the semi-final match against South Africa at Eden Park.[18] He was wicketless in the match but against Australia in the final at Melbourne, Henry showed serious pace[citation needed] and took the two wickets of David Warner and Michael Clarke.[19]
He batted with Jimmy Neesham to set the record for what is, as of March 2018, the highest 9th wicket partnership for New Zealand in ODIs, adding 84 runs as a pair against India in 2016.[20]
Henry's Test debut for New Zealand came during the tour against England in May 2015.[21]
In March 2023, Henry was named in New Zealand's ODI and T20I squad for their tour to Pakistan.[29] On 14 April 2023, in the first T20I, he became the fourth New Zealand cricketer to take a hat-trick in T20Is.[30]