The men aged 24 and 34 were found on Highway 75 just north of Letellier, Man., on the morning of Christmas Eve after walking in the cold for several hours.
A truck driver, who has not come forward publicly, stopped to help the men.
The driver called 911 and an ambulance took the men to the hospital in the town of Morris, Man., before rushing the pair to the Winnipeg Health Sciences Centre.
It's not clear how the men were able to make it over the border without being stopped, but both were disoriented and asking where immigration was when the truck driver stopped.
The 24-year-old man spoke to CBC on the phone on Tuesday and said he is happy to be in Canada.
"They almost froze to death," said Ghezae Hagos, a refugee claim counsellor at Welcome Place, a home for refugees in Winnipeg.
"It's a very shocking sight really," Hagos said, adding the 24-year-old man will likely lose his fingers and had blood seeping through bandages when he visited him Wednesday.
"He didn't know the conditions of Canada."
Environment Canada records show the temperature in the Emerson, Man. area the morning of Christmas Eve was hovering around the -18 mark before the windchill. CBC metrologist John Sauder said skin can freeze within 30 minutes in that temperature.
Despite the two men's ordeal, people that have visited them say they are in good spirits.
Newcomer advocates who spoke to CBC on Thursday say they're seeing an increase in refugees coming to the Manitoba border for help.
"Just in November alone we had 31 claimants," said Rita Chahal, the executive director of the Manitoba Interfaith Immigration Council.
Chahal said there's a network in Minneapolis of refugees who are fleeing the United States.
They are primarily from Somalia and Ghana, she said.
"They obviously have fear (for) their lives and that's why they're asking us in Canada to help them with the protection."
'They almost froze to death': Refugees frostbitten after walking to Manitoba border - Manitoba - CBC News