There is some confusion in the original posting. The question asked for a subring $S$ of $R=\mathbb{Z}_n$ that included the identity. Through the comments, it appears that the OP wanted the additive identity to be included in the subring, not the multiplicative identity. I will provide discussion of both ideas here.
If $S$ must contain the multiplicative identity, $1$, then $S$ contains all sums of $1$ with itself, in other words, all the elements of $R$. Hence, if $1\in S$, then $S=R$. The only way for $S\simeq\mathbb{Z}_2$ is if $R=\mathbb{Z}_2$.
If $S$ does not need to contain the multiplicative identity, then for $S$ to be isomorphic to $\mathbb{Z}_2$, then $S$ must have two elements. It has the $0$ of $R$ because $S$ is a ring and it has one other element, which we'll call $g\not=0$. Since $S\simeq\mathbb{Z}_2$ and $S$ is a subring of $\mathbb{Z}_n$, it must he that $g+g\equiv 0\pmod n$ and $g\cdot g\equiv g\pmod n$.
Since $g+g\equiv 0\pmod n$ and $g\not=0$, we know that $n\mid 2g$ and that $2\mid n$. Therefore, $n=2k$ for some $k$. Assuming, without loss of generality that $0<g<n$, then $2g<2n$, so the only way for $n$ to divide $2g$ is if $n=2g$. In this case, we know that $g\equiv k=\frac{n}{2}\pmod n$.
Since $g\cdot g\equiv g\pmod n$, $g\not=0$, $n=2k$, and $g\equiv k\pmod n$, we consider $k^2$. If $k$ is even, then $n=2k\mid k^2$, which cannot happen because $g\cdot g\equiv g\not\equiv 0\pmod n$. Therefore, $k$ is odd. Since $k$ is odd, $k=2l+1$ for some integer $l$, then $k^2=k(2l+1)=2kl+k=nl+k\equiv k\pmod n$. Therefore, when $k$ is odd, $k^2\equiv k\pmod n$.
Combining all of this, we have that when $n=2k$ where $k$ is odd. $R=\mathbb{Z}_n$ has the subring $S=\{0,k\}$, which is isomorphic to $\mathbb{Z}_2$.
There are several definitions running around, confusing people like you when you read from different sources. For some of them the answer is "Yes", for some the answer is "no".
Some definitions of unital rings wouldn't consider $\{0\}$ a ring at all, since they require that the multiplicative identity is distinct from the additive one. Disregarding that, a subring is a subring by virtue of the inclusion homomorphism, and some sources require that ring homomorphisms preserve the multiplicative identity. In that case, $\{0\}$ is all by itself without any homomorphisms of any kind to any other ring, so it's not included in any other ring either. However, that's usually unporoblematic since these two definitions often come together (so if your definition of homomorphisms would make $\{0\}$ sit alone, your definition of ring would make it not a ring at all).
So, if we let $\{0\}$ be a ring, and we don't require homomorphisms to treat the multiplicative identity with any more care than any other element, then yes, you can include $\{0\}$ into bigger rings and have it be a subring. The same way you can include $\Bbb Z$ as the first component of $\Bbb Z\times \Bbb Z$, for instance.
When I first learned about rings, I prefered the more relaxed requirements, because I thought they made life easier. These days I prefer the stricter requirements because I feel that they make life easier (knowing that the image of $1$ is $1$, and not just any idempotent element really helps some arguments, for instance).
Note that if your definition of rings requires a multiplicative identity to exist in any ring, then you will most likely be in the stricter domain (requiring $1$ to exist means it is nice if we require homomorphisms to respect it). The more relaxed domain typically doesn't require a multiplicative identity to exist in rings (although, of course, some rings happen to have one).
Best Answer
Yes, you're right: the version of the subring test found in the wikipedia article on "subring" was faulty, whereas the article subring test has a correct statement.
I just edited the first wikipedia article to read as follows:
"The subring test states that for any ring R, a subset of R is a subring if it contains the multiplicative identity of R and is closed under subtraction and multiplication.'
I hope all will agree that this is an appropriate statement.
There is a slight discrepancy in that the article on subrings explicitly assumes we are working in the category of rings (in which we have a multiplicative identity which all the homomorphisms much respect), whereas the article on the subring test works in the category of rngs (i.e., there may not be a multiplicative identity and even if there is it need not be preserved by homomorphisms). In the category of rngs, one should state the nonemptiness explicitly, whereas in the category of rings it is guaranteed by the presence of the multiplicative identity.
If anyone has further ideas for improving either of these two articles, please let me know. Or rather, please go ahead and implement them -- be bold, as they say on that other site -- but it would be nice to come back here and tell us what you've done.