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Is there a way to prove that if n = (a^2 + b^2) - (c^2 + d^2), then for every natural n, there are infinitely many a, b, c and d?

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Sure. Demand $a \geq b \geq 0$ as well as $c,d \geq 0.$ Then map $$ (a,b,c,d) \mapsto (25a+11b+24c+13d, 11a-b+11c, 24a+11b+23c+13d, 13a + 13 c+d) $$

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