it came into the fluffy pony community the same way "plot" and the like became MLP slang: 4chan.

hugbox was initially slang used derogatorily in much the same way as "safe space" and "echo chamber". essentially it was a term used to refer to communities who were seen as, wholey/aggressively "anti-negativeity".

not sure exactly how it came to be used as the cuter/happier genre of fluffy pony; it's pretty much unrecognizable...
I could see someone misinterpreting it to just mean "happy stuff" in general, later using it when talking about some fluffy content at which point it just stuck. there was a lot of board-hopping that when the community largely resided on 4chan. mods on /mlp/ hated fluffy pony, and mods elsewhere hated everything related to MLP (fluffy pony included). perfect environment for picking up slang you don't understand and misusing it.

and then "-box" eventually just became a suffix for genres of fluffy pony content.