The man was arrested on Wednesday on suspicion of abducting the 21-year-old philosophy student.
On Friday, Humberside Police said a senior officer had granted permission for further detention of the man in custody.
A spokeswoman said Miss Squire has still not been found.
She was reported missing from the city in the early hours of February 1 and an intensive search has been mounted by police and volunteers over the last week.
On Thursday night, a week since Miss Squire disappeared, police handed out leaflets and drove a large screen featuring images of the missing student around the city centre.
Police said more than 50 student volunteers had joined their efforts by helping to give out flyers.
The address where the man was arrested is about half a mile from Miss Squire's home.
Officers have said Miss Squire was dropped off in a taxi near her home on Wellesley Avenue at 11.29pm on
Thursday January 31 and was seen 10 minutes later on CCTV near a bench on Beverley Road, where a motorist stopped to offer help. She is believed to have been in this area until about 12.09am on Friday.
Police said the taxi driver and motorist have been spoken to and are not suspects.
On Wednesday, Miss Squire's parents Lisa and Russ issued a statement which said: "We'd like to thank everyone who has helped this week in trying to find Libby.
"We have been overwhelmed with people's kindness and support and it's been incredible how all her friends, students at the university, and members of the public have joined together to help us look for her."
Humberside Police said Mr and Mrs Squire, from High Wycombe, Buckinghamshire, had met their daughter's friends and university staff.
In their statement, the couple said Miss Squire - whose full name is Liberty Anna - has two younger sisters and a younger brother, who all live in High Wycombe.