The family's attorney, Bridget Cambria, of Aldea, the People's Justice Center, said the couple had been driving south of Vancouver on 3 October when they took a detour to avoid an animal on the road.
The family said they did not realise they had strayed over the US border.
They were stopped by a police officer who did not read them their rights, nor allow them to "simply turn around" and go back to Canada, according to the complaint.
At first the young family said they were separated - with David Connors being held in a male-only cell, and Eileen Connors and their infant son in a women's cell.
Later, the husband was taken to a detention centre in Tacoma, Washington, while his wife and their baby son were taken to a budget hotel near Seattle-Tacoma International Airport, according to the complaint.
The following morning they say they were driven to the airport, which raised their hopes that they were being flown back to Canada or the UK.
"But that was not the case," Eileen Connors said in a sworn statement. Instead, they were flown to Pennsylvania - on the other side of the country.
They were taken on 5 October to Berks Family Residential Center (BFRC), one of three immigration detention centres in the US that can accommodate families.
US Immigration and Customs Enforcement (ICE) have confirmed the family was in detention at the BFRC facility in Leesport, Pennsylvania.
A spokesman for the agency told the BBC that BFRC "provides a safe and humane environment for families as they go through the immigration process".
"Reports of abuse or inhumane conditions at BFRC are unequivocally false," he added.
The couple's sworn statement said the cells were "frigid", and staff had refused to turn the heating on until the end of next month.
"When I ask how I am supposed to keep my baby warm in this horrible cold, all they tell me is to put a hat on him," Mrs Connors said in the statement.
"My baby can't wear a hat all the time, he feels uncomfortable with hats and mittens and starts to cry."
Staff, she added, confiscated her son's formula for three days, as well as his teething powder, and would only provide "disgusting" blankets that smelled "like a dead dog".
They said the baby's skin was now rough and blotchy and he appeared to have an eye infection.
"We have been treated like criminals here, stripped of our rights, and lied to," Mrs Connors said. "It is not right.
"We have been treated unfairly from day one. It is undoubtedly the worst experience we have ever lived through.
"We have been traumatised and it has even damaged our relationship. No one should have to suffer this kind of treatment."
He knows nothing and thinks he knows everything. That points clearly to a political career.