There has never been any badgers in Badger, or anywhere in Newfoundland for that matter.
I recently saw an article on CBC about how Quidi Vidi Village got its name (LINK). It got me thinking that there is a similar case in Central Newfoundland.
Badger draws its name from Badger Brook. The problem is, there are no badgers any where near Newfoundland. So it wasn’t named for the animal. One story also relates that an old Mi’kmaq trapper (chances are he was a member of the Paul, Barrington, Joe, Beaton or John family) camped near the brook because he was “as tired as a badger.” Another story relates that a similar trapped caught a badger nearby. both of these stories are unlikely as the range of the American Badger is a fair ways away from the ancestral home of the Mi’kmaq. There is an off chance that a otter was mistaken for a badger, who knows, chances are though these old time trappers knew what otters were and didn’t know what badgers were.

Another story is that the name is drawn from the logging foremen or bosses “badgering” the loggers in the early logging camps in the area. Beginning in the late 1890’s there were logging camps in the area. however, the brook was known as Badger Brook before there were any loggers frequenting the area. Logging only really took hold in the area around 1894 when the railway reached the area and the brook became a rail-head to supply the camps.

The name most likely is a result of a misreading of a map. Two hundred and fifty years ago Lieutenant John Cartwright set out to explore the Exploits River. Over the course of six days he traveled down the river naming many of the features including most of the river’s tributaries. One of the larger tributaries appears on his subsequent map as Ranger’s River. It just so happens that Ranger’s River is where Badger Brook is today. Link to map

Cartwright’s map was the only comprehensive map of the area for many many years. I am not sure how widely available it was but I have a confident sneaky suspicion that at some point in the 19th Century the R in Ranger was read as a B and the n somehow as a d, not at all impossible when reading the script in which the map was labelled.
Now as for where the name Ranger comes from, there is only speculation on my part. Badger Brook was once thought to flow out to Badger Bay and so both may bear the misnomer and one may have begat the other. It could have coe from the early exploreres or furriers who rangered far into the country or from a ship. The British Royal Navy had a number of ships over the years that bore the name HMS Ranger, and one of those may have visited Badger Bay, though I cannot find any record of such an event.
I may not be right, but I don’t think the name of the Town of Badger has anything to do with that ornery cousin of the otter.
Further Reading
Interior Nomenclature: The Changing Map and Place Names of Central Newfoundland