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Squire was meant to make his Super Rugby season debut yesterday, off the bench against the Jaguares, but he was a late withdrawal as the Highlanders claimed a 32-27 victory.
Highlanders coach Aaron Mauger explained that Squire's absence was due to personal matters.
"Definitely Liam was a bit of a blow ... he's just dealing with some personal family issues at the moment.
"He won't come back on tour with us - at least for the first week, but as it stands it's likely to be the duration of the tour.
"It's important that our boys are in a space to play the games and at the moment Liam's got to attend to a few things that he needs to deal to."
Squire has yet to play for the Highlanders in 2019, after first having a hip injury that kept him out for the opening month. Just as he was back into full training, he tore the medial ligament in his knee, and now the earliest he will be able to return is in the Highlanders' penultimate game of the regular season.
It's not the only All Blacks blow the Highlanders are going to suffer from, with Jackson Hemopo also not available to play in their games against the Lions and the Stormers due to a knee injury.
"We need to take fresh bodies, he won't be available for the first game against the Lions, so we want to make sure we've got fully fit bodies on the field," said Mauger.
Winger Tevita Li is also a doubt for the touring side after suffering a hamstring injury in the win yesterday, but he will be evaluated before the squad is named this morning.
In better news for the Highlanders, All Blacks winger Waisake Naholo will be included in the 27-man squad after recovering from an ACL injury, while Teihorangi Walden will also return to the squad and be available for selection.