Square, the payments company that makes it easy for small businesses to accept credit card payments wherever you go, is finally catching up with the times today by updating its tiny card reader.

LAS VEGAS — Square sells its own cash register replacement for the iPad, but the mobile payments company has yet to debut any iPhone-compatible accessories for its card reader. Well, until now.

When I interviewed Square CEO Jack Dorsey last December for our 2014 User’s Guide issue, he told me that the company’s payment products, such as the Square Reader credit-card swiper, Square Register ...