I'm not a native speaker, but I will try to answer your question. If I make a misstep here I hope others will correct me, but here is my sense.There are lots of grammar principles of -나.
Could you tell us which usage is it in this case?
Avant Gardener, you nailed itI'm not a native speaker, but I will try to answer your question. If I make a misstep here I hope others will correct me, but here is my sense.
(이)나 has a couple affective qualities depending on how it is being used. When used in a suggesting sentence, it tends to sound friendly or casual (e.g. '맥주나 한잔 할까' - here the addition of -나 doesn't just mean 'or something,' it also has the effect of softening the tone of the sentence).
When used in a command sentence, (이)나 has the opposite effect, and tends to add a sense of anger or pique. Frequently this usage emphasizes that the listener should do the suggested action in contrast to whatever they are doing now. We see this in idiomatic phrases such as '너나 잘해' (meaning mind your own business, more awkwardly translatable in a literal sense as do you, not me). Note that in these cases (이)나 does not literally mean 'or' in the English sense but more communicates frustration or impatience.
This phrasing can be interpreted as rude (when used in combination with insulting or aggressive language it can even carry the nuance of "Shut up and...") and it should be used with caution. That being said, it's a very common grammar pattern in Korean.