= Solution
Let $\xi$ have real rank $r$, choose a bundle metric, and write $D(\xi),S(\xi)$ for its disk and sphere bundles. In a <multiplicative generalized cohomology theory>, a <Thom class in a generalized cohomology theory> is a class
$$
u_h(\xi)\in h^r(D(\xi),S(\xi))
$$
whose restriction to each fiber pair $(D^r,S^{r-1})$ is the suspension of the coefficient unit $1\in h^0(\mathrm{pt})$ under the chosen oriented identification. Equivalently, it is a fiberwise generator over the coefficient ring compatible with the $h$-orientation. <Cup product> with this class gives the <Thom isomorphism theorem>
$$
h^k(X)\longrightarrow h^{k+r}(D(\xi),S(\xi)),\qquad
a\longmapsto\pi^*a\smile u_h(\xi),
$$
where $X$ is the base.
Let $z:X\to D(\xi)$ be the <zero section> and let $\jmath:h^r(D(\xi),S(\xi))\to h^r(D(\xi))$ forget relative supports. The <Euler class in a generalized cohomology theory> is
$$
\boxed{e_h(\xi)=z^*\jmath(u_h(\xi))\in h^r(X).}
$$
Equivalently, identify relative cohomology with <reduced cohomology> of the <Thom space> and pull its <Thom class> back along the <zero section> followed by the quotient map.
Suppose $s$ is a nowhere-zero section. Normalize it to $\widehat s(x)=s(x)/\|s(x)\|\in S(\xi)_x$. The maps
$$
H_t(x)=t\widehat s(x),\qquad 0\leq t\leq1,
$$
give a <homotopy> within $D(\xi)$ from $z$ to the sphere-valued section $\widehat s$. The image $\jmath(u_h(\xi))$ restricts to zero on $S(\xi)$, by the exact sequence of the pair. <Homotopy> invariance therefore yields
$$
z^*\jmath(u_h(\xi))
=\widehat s^{\,*}\jmath(u_h(\xi))=0.
$$
Thus <a nowhere-zero section annihilates generalized Euler classes>:
$$
\boxed{e_h(\xi)=0.}
$$
The argument uses the actual nowhere-zero section and is valid for any chosen Thom orientation in the multiplicative theory.
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