If $E$ is an elliptic curve over $\mathbb{F}_p$ where $p\geq5$ and $\#E(\mathbb{F}_p)$ is even, then does $E$ have a non-trivial $2$-torsion point defined over $E(\mathbb{F}_p)$?
In other words, if we write $E$ as $y^2=f(x)$, then is $f$ reducible over $\mathbb{F}_p$?
Yes, this is Cauchy's theorem (in fact its even less than this as $E(\mathbf F_p)$ is an abelian group). If two divides the order of the group then there is an element of order 2.
Another way to look at it is that the hyperelliptic involution is the map on eliptic curves of the form $y^2 = f(x)$.
$$E(\mathbf F_p) \to E(\mathbf F_p)$$ defined away from infinity as $$ (x, y) \mapsto (x,-y)$$ and it has the point at infinity as a fixed point, so if the cardinality of $E(\mathbf F_p)$ is even then there must in fact be another fixed point, which will be of the form $(x_0,0)$, i.e. a root $x_0$ of $f$.