Solution:

Completing the square, $x^2 + 84x + 2008 = (x + 42)^2 + 244,$ so $y^2 - (x + 42)^2 = 244,$ which factors as $$(y - x - 42)(y + x + 42) = 244 = 2^2 \cdot 61.$$ The two factors have the same parity, and their product is even, so both are even: $y - x - 42 = 2$ and $y + x + 42 = 122.$

Adding gives $y = 62,$ and then $x = 18;$ indeed $18^2 + 84 \cdot 18 + 2008 = 3844 = 62^2.$ Therefore $x + y = 18 + 62 = 80.$