$ (2a + 1)^2 + (2b + 1)^2 + (2c + 1)^2 =\\ (a - c)^2 + (b - c)^2 + (c - a)^2 + (a + b + 1)^2 + (b + c + 1)^2 + (c + a + 1)^2$. $ |a - c|$, $ |b - c|$ and $ |c - a|$ are all positive as we are given that $ a,b,c$ are distinct.

Every integer $ >169$ can be written as a sum of already five nonzero squares, thus checking the rest is what has to be done