As a student of physics one will, on several occasions, indubitably hear professors or other physicists (here is an example, from Physics.SE's highest-rep user John Rennie) tell the famous story that Paul Dirac came upon his relativistic wave equation for the electron, now called the Dirac equation, by "trying to take the square root of the Klein-Gordon operator".
The story comes in many different versions, some dressing it up in a funny anecdote about Dirac, when asked what he was working on, casually replying with "Oh, just trying to take the square root of something". Others simply claim that this was the train of thought he followed in his derivation. Wikipedia has another version, which features Dirac pondering by the fireplace...
A quick look at the first few sections of the paper where Dirac introduced his wave equation, however, shows no trace of "taking square roots" of any kind. Of course, it is clear that Dirac was looking for an operator that was linear---as opposed to the quadratic Klein-Gordon equation---but that doesn't really justify saying that he was looking to take a square root.
Nonetheless, this doesn't really debunk the entire story, since it seems quite reasonable to assume that Dirac may have streamlined his derivation considerably before publishing. Thus, the intuitive idea (maybe he thought it would sound a little silly to publish something so vague-sounding) may have been carefully hidden behind a more rigorous derivation and justification.
My question is: What is this story about "taking the square root of the Klein-Gordon operator" based on? Is there any evidence that Dirac was indeed thinking along these lines? If not, can the story be traced back to any other source?