As you didn't show your results, I'll redo the computation.
Let us substitue the RHS of the two middle equations in the fourth, squared:
$$d^2(x^2+b)y^2=(y^2+c)x^2,$$ or
$$(d^2b-x^2)y^2=(c-d^2)x^2.$$
Then, from the first equation,
$$y^2=(a-x)^2$$
and finally, eliminating $y$,
$$(c-d^2)x^2=(d^2b-x^2)(a-x)^2.$$
This confirms that the problem is of the quartic degree and has an analytical solution, but the formulas are terrible and hardly manageable by hand.
And as the polynomial has three independent coefficients, there is no reason that any simplification occurs and you won't find any simpler analytical resolution (otherwise, you would revolutionize the Galoi's theory).
The other option is to resort to numerical methods, but whether this is feasible by hand is unsure, and it is only doable for particular values of the parameters $a,b,c,d$.
Best Answer
Add $x^2 - y^2 = z$ and $y^2 - z^2 = x$ to get $x^2 - z^2 = x + z$. This means either $x + z = 0$, or $x - z = 1$. Similarly, either $y + x = 0$ or $y - x = 1$, and either $z + y = 0$ or $z - y = 1$.
Now there are a total of $8$ different combinations of equations to deal with.
And then you just proceed to solve for $x$, $y$, and $z$, and find which cases make sense and which ones don't.