"Anything is fine."
Here, (이)나 is added to a question word, so it acts as a generalizer: any/every.
In other contexts, it can also mean
or (comparing two patterns)
It still wouldn't be the same meaning even if its very similar in this particular case.
Also 다 is mostly used as an adverb (아무거나 is a noun) which would also make the sentence not make sense. Actually now that i think about it, this 2nd part may not be true, best to let more people weigh in.