First, let us list the statement "If a quadrilateral is a square, then it is a rectangle" as a statement of the form "If $p$, then $q$". In this case, $p$ is "a quadrilateral is a square", and $q$ is "it is a rectangle".
The converse is then: "If $q$, then $p$". Plugging in, we get "If a quadrilateral is a rectangle, then it is a square". However, this is obviously false, as a rectangle does not have to have four sides of the same measure.
The inverse is then: "If not $p$, then not $q$". Plugging in, we get "If a quadrilateral is not a square, then it is not a rectangle". This is also false, since a quadrilateral can be a rectangle but not be a square. 
Therefore, the answer is $\boxed{\textbf{D}}$.
Given the true statement: If a quadrilateral is a square, then it is a rectangle. 
It follows that, of the converse and the inverse of this true statement is:
$\textbf{(A)}\ \text{only the converse is true} \qquad \\ \textbf{(B)}\ \text{only the inverse is true }\qquad  \\ \textbf{(C)}\ \text{both are true} \qquad \\ \textbf{(D)}\ \text{neither is true} \qquad \\ \textbf{(E)}\ \text{the inverse is true, but the converse is sometimes true}$