Home Repair of Refrigerator Waterline
You need a plastic quick coupling.  Go to a professional plumbing supplier and buy a 5/16 x 5/16 Parker fitting. These are similar to Sharkbite fittings, but plastic and about half the size of the smallest Sharkbite. They come in 1/4, 5/16, 3/8 and 7/16.  A number of companies such as Celcon and John Guest make them, but Parker invented them so like Hoover for a vacuum, they are called 'Parkers'. 
Simply cut your pipe right at the leak with a very sharp craft knife then fit each end into the straight fitting.