Clearly, the grasshopper can arrive at the integers $+1$ and $-1$, in one jump. And also, he can arrive at the integer $14$ using three jumps to the right ($1 + 4 + 9 = 14$).
Note the following: if the grasshopper can arrive at a number $a \in \mathbb{Z}$ using $n$ jumps, then he can also arrive at the numbers $a-4$ and $a+4$ using $n+4$ jumps, by jumping left, right, right, left (for $a-4$), and right, left, left, right (for $a+4$), because of
$$a - (n + 1)^2 + (n + 2)^2 + (n + 3)^2 - (n + 4)^2 = a - 4,$$
$$a + (n + 1)^2 - (n + 2)^2 - (n + 3)^2 + (n + 4)^2 = a + 4.$$
Since the numbers $0$, $+1$, $-1$ and $14$ all have distinct remainders modulo four, the grasshopper can arrive at each integer in a finite number of jumps.