You are not alone in finding this problem a surprisingly tricky one.

Surprisingly little has been written about the origins of this term, and even when it was current the sorts of sources one would expect to illuminate its meaning are brief and to-the-point rather than usefully discursive. Thus Shelton's The Historical Finger-Post (1879) merely refers to the term as an "eccentric title", and even Hobson-Jobson, the renowned dictionary of Anglo-Indian terms, suggests the words were simply "an old personification of the East Indian Company," and shows more interest in the fact that they caused considerable confusion since they were "by the natives often taken seriously, and so used, in former days." The dictionary goes on to quote an unidentified Indian who, in 1826, "said that according to some accounts, he had heard the Company was an old Englishwoman.... then again he told me that some of the Topee wallas say 'John Company,' and he knew that John was a man's name, for his master was called John Brice, but he could not say to a certainty whether 'Company' was a man's or a woman's name."

The issue was, I think, probably less baffling to the British servants of the company. The East India Company was the country's senior joint stock company, and certainly by the middle of the 18th century it was also the wealthiest and the most powerful – it was the pre-eminent British company of the time, and set standards against which others measured themselves. "John", similarly, was the dominant Christian name of the period, so just as the British of the First World War period termed German soldiers "Fritz" because Fritz was considered a typical, and characteristic, German name, so the quintessentially British company of 150 years earlier was naturally nicknamed "John".

The term, it's worth adding in conclusion, didn't actually originate with the EIC, though – it was borrowed from the Company's Dutch equivalent, the VOC, which was known for exactly similar reasons as "Jan Compagnie" from the first half of the 17th century.