With $p$ a prime integer, I am interested in showing the existence of four non-negative integers $\alpha_0,\alpha_1,\alpha_2,\alpha_3$, not all zero, such that $$\alpha_0^2+\alpha_1^2+\alpha_2^2+\alpha_3^2\equiv 0\mod p.$$ I am, of course, aware of the Lagrange four square theorem of which this is a special case. However, the fact that I am concerned with only prime $p$, and also that I have the looser modulo condition, leads me to hope that there is a simpler, more direct way of proving this.
My thoughts so far have been directed towards trying to tie quadratic residues into some sort of pigeonhole principle argument. That hasn't panned out. This question is motivated by a (double-star) problem in Herstein's Topics in Algebra. Ideally I am looking for hints rather than anything fleshed out. Thanks in advance.