It's all in the correct reference. Cassels, Rational Quadratic Forms, chapter 9 "Integral Forms over the Rational Integers," pages 163-164, Examples 9-11. Example 11(i) says that, for "odd" matrices, we can cut down the dimension by 2 and write $y_1^2 - y_2^2 + g(z_1, \ldots , z_{n-2}).$ The determinant of $g$ is still $\pm 1,$ so the only problem is that $g$ may be "even."
Next, if $f$ is "even" the quadratic form can, in fact, be written $ 2y_1 y_2 + g(z_1, \ldots , z_{n-2}).$
So, all we really need is to show, as in Sylvester's Law of Inertia, that the resulting form $g$ continues to be indefinite. Presumably your condition with $J M J^T = -M$ can do this.
Otherwise, without your $J$ condition, Example 11(vi) says that either $f$ or $-f,$ if "even," is equivalent to a sum of some $2x_j y_j$ terms along with a single $\mathbb E_8$ lattice. CASSELS
I was uneasy about the possible need to mix 2 by 2 blocks of both types, despite Hahn's statement, but $$ \left( \begin{array}{cccc} 3 & 4 & 2 & 2 \\\ 2 & 3 & 1 & 2 \\\ 0 & 1 & 1 & 1 \\\ 2 & 3 & 2 & 1 \end{array} \right) \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\\ 0 & -1 & 0 & 0 \\\ 0 & 0 & 0 & 1 \\\ 0 & 0 & 1 & 0 \end{array} \right) \left( \begin{array}{cccc} 3 & 2 & 0 & 2 \\\ 4 & 3 & 1 & 3 \\\ 2 & 1 & 1 & 2 \\\ 2 & 2 & 1 & 1 \end{array} \right) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\\ 0 & -1 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & -1 \end{array} \right) $$