Believe it or not, the basic mechanisms of the modern "flushing toilet" were invented in 1596 by Sir John Harington, and was called the Ajax. Sir John installed one in the palace of Elizabeth I of England, but it's rumored that she did not like using it because it was too loud.
A variety of designers and inventors tweaked the design for the next 300ish years before plumbing companies like Thomas Crapper's really popularized the technology and flushing toilets started appearing more in private homes.
It makes one wonder why took so long for flushing toilets to become a household staple. I mean, I know it was probably due to some cost or mass production issue, but I feel like I would have made it a priority! :)
Onto the obvious questions:
1. Did the nickname "john" for a toilet come from Sir John Harington?
Not that I can see. Toilets (flushing or non) began being called "johns" in the mid-1800s, probably because, at that time, some people referred to outhouses as "johnny houses". I don't imagine these people had access to the full history of the toilet to know who Sir John even was!
2. Well, did the nickname "crapper" for a toilet come from Thomas Crapper?
No, says snopes.com. The word "crap" actually originated around 1845, which was about 35 years before TC went into the toilet industry. Maybe his name inspired his profession?