Take all cases When a,b,c are odd When a is even , b,c are odd When a,b are even and c is odd When a,b,c are even

For integrs $q_1,\ q_2,\ q_3$, let $a=2q_1+r_1,\ b=2q_2+r_2,\ c=2q_3+r_3\ (r_1,\ r_2,\ r_3=0,\ 1)$, we have $a^2+b^2+c^2=4(q_1^2+q_2^2+q_3^2+q_1r_1+q_2r_2+q_3r_3)+r_1^2+r_2^2+r_3^2$, thus $4|a^2+b^2+c^2\Longleftrightarrow r_1^2+r_2^2+r_3^2=0$, yielding $r_1=r_2=r_3=0$, or $a=2q_1,\ b=2q_2,\ c=2q_3.$