Dirichlet showed that if q is a prime number and q == 3 mod 4 then the sum of the nonresidues of q minus the sum of the quadratic residues of q in the range 1, 2, …, q − 1 is an odd multiple of q. What coverage of the odd numbers is given by these odd mulitiples?
These odd multiples are (save for $q=3$) the class numbers $h_{q}$ of the fields $\mathbb{Q}(\sqrt{q})$. (This follows from the analytic class number formula,) By the BrauerSiegel theorem, $\log h_{q}\sim\log\sqrt{q}$. From this it looks plausible that all possible odd $h$ could occur. I'm not aware of any proof that this is the case (nor of any potential counterexample). 


It is very likely that every (positive) odd number is covered by a sum of this type. As Robin Chapman points out, this is equivalent to asking, for a given odd number $h$, whether there exists an odd prime $q\equiv3\pmod{4}$ such that $\mathbb{Q}(\sqrt{q})$ has class number $h$. Let $N(h)$ be the number of quadratic imaginary number fields with class number $h$ (for odd $h$)  note that such a field is already necessarily of the form $\mathbb{Q}(\sqrt{q})$ for $q\equiv3\pmod{4}$ by genus theory. This rephrases your question as "Is $N(h)>0$ for all odd $h$"? By the calculation of mark Watkins mentioned in Stoppie's answer, it is known that $N(h)>0$ for all $h\leq 100$. More importantly, $N(h)$ gets rather large  $N(1)=9$ is HeegnerStark, and this is the smallest value of $N(h)$ in this range. For instance, $N(h)>100$ for all $h>37$, and $N(99)=289$. So we'd like an assurance of some sort of a rough upward trend to guarantee that $N(h)$ does not hit zero at some point down the road. Enter Soundararajan's article "The number of imaginary quadratic number fields with a given class number", which studies asymptotics of $N(h)$. Soundararajan remarks that $N(h)$ should be on the order of $h$, and conjectures more precisely that $$ \frac{h}{\log h}\ll N(h)\ll h\log h. $$ This is accompanied by a probabilistic argument as to why this is a reasonable conjecture. I have not worked through the analysis, but it seems likely to me that if one's only goal was to prove that $N(h)>0$ for all $h$ (a lower bound was not a goal of the paper), this heuristic argument could be made sufficiently rigorous, especially when combined with the data for $h<100$ above, to rule out pathologies for small inputs. 

