This is the first time I've ever heard such a thing. IF it's true it would have been in his youth or his very early adulthood prior to becoming a circuit court lawyer. Now there may be a couple reasons why people say this if it is not true.
First is that bare-knuckle boxing was common place in Lincoln's lifetime. Now boxing gloves did exist in the ancient world, though ancient boxing wasn't quite like modern. For instance, boxing gloves could actually be a weapon in the ancient sport. The Roman cestus was basically an ancient boxing glove that could have a metal plate, spikes, and blades incorporated into it to do some serious damage. Boxing was largely banned in the 4thcentury AD and remained so until the 17th century when it really began to resurface in England. Gloves weren't a common place sight in boxing matches during that time but they weren't unheard of either. It wasn't until the Marquess of Queensbury rules were introduced that gloves became a commonplace sight in boxing matches. And they weren't introduced until 1867 in England and 1889 in the US. So the chances tended to be good that anyone who fought did so bare knuckled.
Second is that Lincoln actually was challenged once to a duel. Cormac O'Brien discusses in his book "Secret Lives of the Civil War" how in the early 1840s Illinois state auditor challenged Lincoln to a duel after Lincoln, under the pen name Rebecca, had written vicious letters to the editor against him, which his future wife and a friend of hers made worse by further publishing under the same pen name. Being the challenged party Lincoln had the choice of weapons. He decided on something he'd have an advantage with based on his height and reach, broadswords. The duel didn't end up happening as their seconds managed to cool them both down. Douglas Lee Gibboney in his "Scandals of the Civil War" also discusses this duel and places it in 1842. He also states that Lincoln and Shields became friends after this and that Lincoln made him a general in 1861.
With an instance of nearly fighting a duel and bare knuckle boxing being common place it would not be shocking folks would claim he was a brawler whether or not he actually was. But a comment by Lincoln himself seems to suggest otherwise. According to O'Brien's book, Lincoln once said "Doesn't it strike you as queer that I, who couldn't take the head off a chicken and who was sick at the sight of blood, should be cast into the middle of a great war, with blood flowing all around me?"