It means everything is important. In this case it follows the plain logic of the words. "You left out a comma on line 131." "That comma isn't important." "Nothing is unimportant. Why, a missing comma in a contract once cost Roger Communications a million dollars. A missing comma once destroyed a NASA rocket." Your friend is simply wrong. However, it is "a stretch," but it is also _possible_ to read the phrase as meaning that nothingness, itself, can be important. It's almost a joke or a pun or a play on words, but it's possible. "The invention of a symbol meaning 'zero' revolutionized mathematics and arithmetic. A zero may mean "nothing," but sometimes 'nothing' is important."

Yes, "nothing is unimportant" means everything is important.

It can either mean "Everything is important" or "The fact that there is nothing is unimportant."