Northampton University student Nabila Nanfuka, 22, from Neasden, north west London, died following the incident at the Lava & Ignite club on Tuesday night.
Eyewitness James Adetula, 21, from London, said the crush happened when a fire alarm went off, prompting a stampede for the exits.
Northamptonshire Police said two other women were taken to Northampton General Hospital with crush injuries and remained in a critical condition.
Mr Adetula said he found out on Wednesday morning that his friend Ms Nanfuka, who was studying leisure and tourism, was killed in the crush.
He said: "It was a normal event, like a club night, obviously it just ended not the way everyone expected it to end."
Mr Adetula said there was an announcement which sent clubgoers towards the exits.
He said: "It was 'everybody the coaches are leaving, make your way to the exits', so everyone went for the exits. Then next minute the fire alarm went off and everyone started panicking."
He went on: "There was obviously a stampede of people in there when the fire alarm went off."
Describing Ms Nanfuka, Mr Adetula said: "My heart is not settled because the person who passed away was a dear friend. She was a good girl, a really good girl. She wasn't a trouble-maker."