Is it true that $\mathbb{Z}_n$ doesn't contain a proper subring which is a field?
I have a contradiction that shows that this isn't true. Take the ring $\mathbb{Z}_{10}$ and the proper subring $\langle 2 \rangle = \{ 0,2,4,6,8 \}$ which is clearly a field under the same addition and multiplication modulo 10.
But I've come across a statement that says that for a positive integer $n$, $\mathbb{Z}_n$ actually doesn't contain a proper subring which is a field. If it is so, then this should contradict the example that I've given above. How can this be possible? Is there any information that I'm missing?
It depends what you consider a ring (and hence, a subring). Many authors include the identity in the definition of a ring. Under this definition, a subring of a ring must contain the original identity. In this case, ${\bf Z}_n$ has no proper subrings, so if it is not a field itself, it has no subrings which are fields.
Otherwise, if your rings are not necessarily unital, a subring may not contain the original identity, in which case your example is indeed correct (with $6$ as the multiplicative identity). An easier example is just $\{0,5\}$.