We color a point (a,b,c) with white/black if $a+b+c$ is even/odd respectively. The Elf's first move will be at a point $(x,y,z)$ with $x^2+y^2+z^2=n$, and easy to observe that $(x,y,z)$'s color depends on the parity of $n$ (white/black when $n$ is even/odd, as $n=x^2+y^2+z^2=x+y+z( mod 2)$). As (0,0,0) is even, if $n$ is even then the elf only goes through white points, yet she'll never be on a black point. If $n$ is odd, then the elf's points' colors alternate(1st move is black, 2nd white, 3rd black and so on). As she gets normal after an even number of jumps, she will not be on black points while being normal. Thus, there is no such $n$.