Humberside Police said the man was arrested at an address in Raglan Street, Hull, on Wednesday night.
A spokeswoman said 21-year-old Miss Squire has still not been found.
She was reported missing from the city in the early hours of Friday morning and an intensive search has been mounted by police and volunteers over the last week.
The spokeswoman said: “The man remains in our custody, assisting us with our enquires.
“We have not yet found Libby and doing so remains our top priority.
“Libby’s parents have been informed of the man’s arrest and we’re continuing to support them at this difficult time.”