My 2002 World Almanac shows Australia at 2.7 people per square kilometer as of 2001. Only two countries were more sparsely populated: Namibia at 2.2 and Mongolia at 1.7.
The most densely populated country was Monaco, really just the size of a city, at 15,921.0, followed by Singapore at 6,891.7 and Vatican City at 2,200.0, also technically politically independent, but the size of cities.
The most densely populated country which is bigger than New Jersey in area was Bangladesh, at 980.3 people per square kilometer.
My apologies to anybody who is from a country smaller than New Jersey, but that is just too small to be taken seriously. 😀