The standard reference for continued fractions (at least as of 100 years ago) is
Perron, Oskar, Die Lehre von den Kettenbrüchen. Leipzig—Berlin: B. G. Teubner. xiii, 520 S. $(8^\circ)$ (1913). ZBL43.0283.04.
Section 81, Satz 3 states (in modern language): Consider the continued fraction
$$
b_0 + \frac{a_1}{b_1 + \displaystyle\frac{a_2}{b_2+\ddots}}
$$
with $a_n = a$ and $b_n = dn+c$. Provided $a \ne 0, c \ne 0,$ and $d \ne 0$, the value of the continued fraction is
$$
V = \frac{c \; {}_0F_1(;c/d;a/d^2)}{\;{}_0F_1(;1+c/d;a/d^2)\;}
$$
Take $a=-\pi^2$ so $a_n = -\pi^2$ and
$c=2, d=4$ so $b_n = 4n+2$.
The value of the continued fraction is computed as
$$
V = \frac{\;2\;{}_0F_1(;1/2;-\pi^2/16)\;}{{}_0F_1(;3/2;-\pi^2/16)}
$$
But the numerator is $2\cos(2\sqrt{\pi^2/16}\;) = 0$ and the denominator is $2/\pi$, so
$$
V = 0 .
$$