Everybody knows that there are $D_n=n! \left( 1-\frac1{2!}+\frac1{3!}-\cdots+(-1)^{n}\frac1{n!} \right)$ derangements of $\{1,2,\dots,n\}$ and that there are $D_n(q)=(n)_q! \left( 1-\frac{1}{(1)_q!}+\frac1{(2)_q!}-\frac1{(3)_q!}+\cdots+(-1)^{n}\frac1{(n)_q!} \right)$ elements in $\mathrm{GL}(q,n)$ which do not have $1$ as an eigenvalue; here $q$ is a prime-power, $(k)_q!=(1)_q(2)_q\cdots(k)_q$ are the $q$-factorials, and $(k)_q=1+q+q^2+\cdots+q^{k-1}$ are the $q$-numbers.

Now, there are $D_n^+=\tfrac12\bigl(|D_n|-(-1)^n(n-1)\bigr)$ and $D_n^-=\tfrac12\bigl(|D_n|+(-1)^n(n-1))\bigr)$ even and odd derangements of $\{1,2,\dots,n\}$, as one can see, for example, by computing the determinant $\left|  \begin{array}{cccccc}    0 & 1 & 1 & \cdots & 1 & 1 \\    1 & 0 & 1 & \cdots & 1 & 1 \\    1 & 1 & 0 & \cdots & 1 & 1 \\    \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\    1 & 1 & 1 & \cdots & 0 & 1 \\    1 & 1 & 1 & \cdots & 1 & 0  \end{array}  \right|$ and looking at the result.

How should one define $D_n^+(q)$ and $D_n^-(q)$?

In short, I'd tell your friend:  "If you believe a ring can be understood geometrically as functions its spectrum, then modules help you by providing more functions with which to measure and characterize its spectrum."

Elements of a module over a ring $R$ are like generalized functions on $Spec(R)$.  We can talk about the support of a module element, or its vanishing set.  More concretely, think of how global sections of a line bundle can act as functions you can use to define map into projective space.  

When you glue together a module on open sets of a spectrum or a scheme, you get to glue using maps which are module isomorphisms, which are more flexible than the ring isomorphisms required to glue together a scheme.  Borrowing intuition from smooth manifold land, the twist in the Moebius band (as a line bundle on the circle) is formed by gluing a copy of the reals to itself via multiplication by $-1$, a module map, not a ring map.  This allows us to think of functions like $\cos(\theta/2)$ as being globally defined: as a map to the Moebius band. 

In the same vein, when you have a representation $V$ of a group $G$, each element $v\in V$ gives you a nice evaluation map from $G$ into $V$, so lurking everywhere we've got these morphisms from our object of interest into a known object, which are nicely related to each other via the group laws.  A fortiori, this certainly doesn't capture the full utility of group representations, but a priori I think it's a decent justification.
