I read this short story I think as a student c. 1980s. Most of the stuff I read then were older anthologies from second-hand shops and bargain bins.
A man lives in a house which is identical to every other house and drives an identical car to work every day.
One day he chooses to do something rebellious (or just unusual) and automatic shutters built into his house trap him until the authorities turn up to arrest him.
He is taken to an insane asylum where he finds other rule-breakers like himself under treatment. These include a woman.
Eventually the 'patients' all escape from the asylum and they turn into animals. He turns into a unicorn. I think the woman may turn into a unicorn too. They gallop off to freedom.
The word unicorn may appear in the title, which may have been a quotation.