Cheshire police said David Newall, 36, from Bradford, and Rhys Barker, 18, from Castleford, died after the crane collapsed at the building site in Crewe, Cheshire, on Wednesday.
Both men had been working at the site on Dunwoody Way and died at the scene.
Elizabeth Fairhurst, the principal of Mr Barker’s former school Airedale Academy, released a statement.
She said: “We were shocked and saddened to hear of the tragic death of Rhys Barker. Rhys joined Airedale Academy in Y7 and remained with us into the sixth form to study business.
“He was a friendly young man, extremely well-mannered who was popular with other students. His teachers remember him as hardworking, dedicated and quiet; he regularly took part in sporting events for the school, including staff v student competitions.
“The staff and students who knew Rhys have all been affected by this. One of Rhys’ friends has set up a crowd funding page to raise money for a memorial bench in the school grounds and students have organised a balloon release in his memory.
“Our thoughts are with his family and friends at this difficult time.
A 45-year-old man from Glossop remains in a stable condition at the Royal Stoke Hospital.
A force spokesman said post-mortem examinations showed both men died from crush injuries.
An investigation into the circumstances is being carried out by police in conjunction with the Health and Safety Executive.
Emergency services were called to a new build site at 4.30pm on Wednesday following reports of a collapsed crane.
A parent and child in a house damaged by the falling crane were taken to hospital for a check-up as a precaution but were not injured, North West Ambulance Service said.
Speaking on Thursday, Jonathan Seddon, chief executive of Seddon, which runs the construction site, said: “I’m deeply saddened to say that following the incident involving the erection of a crane, two people working on site have lost their lives, while a third remains injured in hospital.
“The crane also struck an adjacent house, the occupants were taken to hospital and have since been discharged.
“On behalf of everyone at Seddon, I’d like to offer my heartfelt condolences to the family and loved ones of those who so tragically lost their lives yesterday. Our thoughts and prayers are with everybody involved, and words cannot express how we feel.”