Nuckle Under is rarely used. It is rarely used. I understand it as - "to prepare mentally for a hard - (not always literally) painful situation. Go down = defeated. Not the same as give in though. It is more like being knocked out or being rendered helpless. Being knocked out is defeat without consent. So, "Will not go down without a fight" means you may or may not go down, but you will fight either way. This is a more common thing to hear. It is said when someone is the underdog or littler person, who wants to let the other know they are committed to the fight and nearly fearless.

Go down, give up, give in and knuckle under have the same meaning when used this way. All are commonly used, but while give up and give in are both formal and informal, go down (unless used literally, i.e., the boxer will go down) and knuckle under are more informal.

I am not going down without a fight = In the end, I may be defeated, but I will fight, nevertheless. "Knuckle under" is an idiom (as opposed to slang) and would not be used in extremely formal literature, but is found in everyday use.