A solution to a Diophantine equation with three unknowns (similar to the Fermat Theorem) I need to either solve the following Diophantine equation with three unknowns:
$n_{1} (n_{1} +1) \pm n_{2} (n_{2} +1) \pm n_{3} (n_{3} +1) = 0$,
where $n_{1,2,3}$ can be positive or negative -- or perhaps prove that this equation doesn't have a solution other than the trivial one
$n_{1} = 0,\hspace{5mm} n_{2} = n_{3}$
plus the obvious permutations.
Given that this equation looks pretty much like the one in the Fermat Theorem, I personally think that the latter is more plausible.


Here is an answer to the question as originally posted. Later on, the OP has somewhat displaced the goalposts.

If $n$ is an integer, no matter what its sign is, then $n(n-1)$ will be either zero or positive, with zero exactly if $n=0$ or $n=1$.
Since neither of the terms in your equation can be negative, the only way for the entire thing to be zero is if each term is zero, so you have exactly $8$ solutions, corresponding to each combination of $n_1,n_2,n_3\in\{0,1\}$.


Answer to the currently (as of this writing) asked question:

The numbers of the form $n(n+1)$ are
$$ 0, 2, 6, 12, 20, 30, \ldots $$
Their differences are the successive even numbers, and since all the numbers are themselves even, we can just pick one of them and express it as the difference of two others, for example
$$ 3(3+1) + 5(5+1) - 6(6+1) = 0 $$
and in general we have solutions of the form
$$ n_1 \in \mathbb Z, \qquad n_3 = \frac{n_1(n_1+1)}2, \qquad n_2 = n_3-1 $$
This doesn't exhaust the solutions; we also have, for example
$$ 5(5+1) + 6(6+1) - 8(8+1) = 0 $$
$$ 9(9+1) + 13(13+1) - 16(16+1) = 0 $$
$$ 11(11+1) + 14(14+1) - 18(18+1) = 0 $$
