E.g. 負けずに＆負けぬように Makezu vs Makenu
Not exactly grammatically. According to Wikipedia, ず is 連用形 while ぬ is either 終止形 or 連体形. In your case if we swap them and obtain 負けぬに (I cannot even directly type this one out with my Japanese input) and 負けずように it won't be grammatically correct. Also note the meaning of your two phrases are different. The first one means "without being defeated" while the second means "so as not to be defeated". The last thing to note is that they are all archaic forms of ない but still commonly used in contemporary Japanese.