Squire joined the NTT DoCoMo Red Hurricanes in Japan on a two-year deal after the 2019 season but later made an early return to New Zealand in September 2020.

After switching Super Rugby teams in 2016, heading south to the Highlanders and becoming a regular replacement for Elliot Dixon and Luke Whitelock off the bench, he was selected on 29 May 2016 for the All Blacks' 32-man squad for the Steinlager series against Wales.

Squire was retained in the 33-man squad for the 2016 Rugby Championship and became a regular back-row and locking cover off the bench for the All Blacks during the competition in tandem with fellow newcomer Ardie Savea.

Squire made a recovery from injury to start for the Highlanders in Round 17 of Super Rugby, taking part in the 40-17 win over the Reds on 14 July.

Squire made three more starts in the 2017 Rugby Championship, supplanting 81-test veteran Jerome Kaino as New Zealand's clear first-choice blindside flanker during the series.

Squire's ongoing injury troubles saw him miss many games for the Highlanders during the 2018 Super Rugby season, but he remained a first-choice player for New Zealand internationally.

The hand injury he suffered against South Africa, in the shocking 34-36 loss, caused him to miss the rest of the Rugby Championship, which New Zealand won once again.

After a poor performance against England, in a slim-margin 16-15 win, Squire was then injured yet again against Ireland, being replaced by Scott Barrett only 32 minutes into New Zealand's 9-16 loss to the Irish.

[11] Squire claimed on social media, that mental health issues, stemming from injuries and the pressure of test rugby, were contributing factors towards his reluctance to play internationally, which saw him gain widespread support and praise from fans and players alike.

[12][13] Squire missed selection for the 2019 Rugby World Cup, but was praised by All Blacks Head Coach, Steve Hansen, for making a "brave call".