The 26-year-old back row forward broke his thumb in the Otago Highlanders' 45-41 victory over the Cheetahs in Bloemfontein on Friday.
He joined fellow loose forwards Kieran Read (thumb) and Jerome Kaino (knee) on the injury list with the start of the Lions tour less than a month away and the first of three tests taking place on June 24.
Hooker Dane Coles is also an injury concern battling concussion symptoms, while outside backs Israel Dagg (knee) and Nehe Milner-Skudder (foot) are also still recovering.
"You always have concerns but at the same time you can't control that," Hansen told Fairfax Media on Saturday.
"All you can do is work away at getting as much information as you can around them and help the players work on their rehab to get better.
"We're sitting in a situation where Reado and JK should both be okay if everything goes right and Coles with his injury now it's got the concussion side to it you've just got to be patient and wait. You can't force that.
"You know you're going to get injuries and we'll get more of them. You have to deal with each situation as it comes along but we don't name the team until June 8 so there's no point panicking."
Squire would have been an almost certainty for the All Blacks squad and likely to have started at blindside flanker or number eight had Read or Kaino not recovered in time.
Hansen, however, said that players with test experience in Elliot Dixon and Steven Luatua were still in contention for loose forward squad spots, as was the uncapped Brad Shields.
"You've got the guys that have been there before like Dixon, Luatua ... Shields is playing well just to name a few. There's a lot of guys putting their hands up.
"As we go a bit further into the competition we'll see what their strengths are like. But we're not panicking."
Both Kaino and Read could face the Lions for their Super Rugby sides - the Auckland Blues and Canterbury Crusaders, respectively. The Blues face the Lions on June 7, while the Crusaders face the tourists in Christchurch on June 10.
The All Blacks have also said they are hoping to organise a test match against Samoa on June 16.
(Reporting by Greg Stutchbury; Editing by Larry Fine/Nick Mulvenney)
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