Why is it 나 instead of 사? Is it like Japanese where there are multiple ways to say 4 since it's a culturally bad number?
나흘 - 4 days
하루(1 day), 이틀(2 days), 사흘(3 days), 나흘(4 days), 닷새(5 days), 엿새(6 days), 이레(7 days), 여드레(8 days), 아흐레(9 days), 열흘(10 days), 열하루(11 days), 열이틀(12 days), ...
And 나 probably comes from the Korean counting system (하나, 둘, 셋, 넷,...).
Otherwise, you're right, Koreans also often avoid putting the number 4 which sounds like "death", e.g. in elevators, where the letter F is used instead.