EDIT: Emil has pointed out that i had an incorrect interpretation of the question. The idea about not using 49 seems to work for all numbers. Evidently it also works for all numbers that can be written as the sum of three squares. Not two squares, though, any prime of the form $49 + x^2$ such as 53 has no alternative, for instance.

ORIGINAL: On page 140 of *The Sensual Quadratic Form* by John Horton Conway, he proves that the positive integers that are not the sum of four **positive** squares are
$$ 1,3,5,9,11,17,29,41, 2 \cdot 4^m, 6 \cdot 4^m, 14 \cdot 4^m. $$
I think this must be what you want, because he specially mentions the bound $49.$
An important detail in this is that, if any multiple of $8$ is the sum of four squares, these squares are even.