O(n) external intersection points?
Consider $n$ axis-parallel squares of equal size shifted along the line $y = x$ such that the bottom-left corners of the squares all fit within the first square. E.g. for $n = 4$:



Then you have $2(n - 1)$ external intersection points but $(n-1)(n-2)$ internal ones (and thus $n(n-1)$ total).