Suppose there exists rational numbers $x,y,z$ such that $$7=x^2+y^2+z^2$$Let $k$ be the smallest positive integer such that $kx,ky,kz$ are all integers. Let $a=kx$, $b=ky$ and $c=kz$. Since $k$ is minimal, this means $\gcd(a,b,c)=1$. We have: $$7k^2=a^2+b^2+c^2$$All perfect squares are either $0,1,4$ modulo $8$. The possible values of $7k^2$ modulo $8$ are $0,4,7$. The possible values of $a^2+b^2+c^2$ modulo $8$ are $0,1,2,3,4,5,6$. This shows that either $a^2+b^2+c^2\equiv0$ or $4$ modulo $8$, which is only possible when all of $a,b,c$ are even. This is a contradiction, because $\gcd(a,b,c)=1$. Hence, $7$ is not spheric.