Having a very undignified and unruly attitude.
Being a rebel or the "American Badass" type of demeanor.
Having a bear tattoo and being an uncaged animal.
That guy has a huge bear tattoo on him. I wonder if he's unruly and rebellious?
-Oh, fuck yeah he is! That is Bear Johns!
(Example) "I think I'm going to be Bear Johns today. I'm going to be unruly and then go home and stretch my sister's asshole out."
A guy who you go out with on a couple of dates and appears to have it together, but then you wake up in the middle of the night and he is cuddling his teddy bear instead of you.
"So how did your date go?"
"Dude, the night was going well until we went back to his room and I saw the teddy bear on his bed. After that I totally had to bounce, I can't end up with a bear john on my hands!"