It's known that any natural number can be expressed as a sum of four squares.
Conjecture:
A natural number can be written as a square sum of four different nonzero integers if and only if it not is a potency of two and not is a member of the set:
{3,5,6,7,9,10,11,12,13,14,15,17,18,19,20,21,22,23,24,25,26,27,28,29,31,33,34,35,36,38,40,41,43,44,47,48,49,52,53,56,59,68,73,76,80,83,88,96,104,112,128,144,160,192,208,224,304,320,384,416,448,640,768,832,896,1280,1536,1664,1792,2560,3072,3328,3584,5120,6144,6656,7168,10240,12288,13312,14336,20480,24576,26624,28672}.
So far tested up to $2^{16}$.
The OEIS entry for the positive integers that are not the sum of four distinct non-zero perfect squares says that your conjecture is wrong. Specifically, their list of the first $1000$ numbers shows a lot of numbers that are not powers of $2$ even after your $28\,672$. Here are the numbers from that list between $2^{15}$ and $2^{16}$ in the sequence: $$ \begin{array}{} 31744\\ 32768\\ 33792\\ 34816\\ 36864\\ 37888\\ 38912\\ 40960\\ 43008\\ 44032\\ 45056\\ 48128\\ 49152\\ 53248\\ 56320\\ 57344\\ 59392\\ 61440 \end{array} $$