More fun in 2016, Part 4

Number Theory Level 5

Find the largest integer $n$ such that $n, n+2016$ , and $n-2016$ are all perfect squares.

Enter 666 if you come to the conclusion that no such $n$ exists.

Bonus Question (very hard): Does there exist a rational number $q$ such that $q,q+2015$ and $q-2015$ are all squares of rational numbers?