I figured it came from a similar expression like "happier than a pig in shit" implying the person is happy about something they really shouldn't be happy about.
If you lived in the country before TV, internet, gaming, or even electricity, you would probably have seen a lot more possums especially when you're hunting them for tomorrow's dinner.
I always figured it was meant to describe the kind of person who could sit and happily eat an entire plate of shit, and yet still think they've got things more figured out than anyone else.
Because when your boss is doling out shit for everyone to digest, everyone sits there with a wide smile on their face to show how impressed they are with boss's fecal dissemination talent.