Amplification Effects: Negative Denominator
$\beta$ appears to be a decision variable, at firm level. Then its value should be determined optimally, under some criterion. So what is the criterion, what is the objective function which should be optimized over the value of $\beta$?

It appears that we should maximize the value of the option $V(\beta)$

$$V(\beta) = \frac{-c + q(\theta)(1-2\beta)J}{\rho - q(\theta)\beta}$$

The first derivative is (I will dispense with $\theta$)

$$\frac {\partial V(\beta)}{\partial \beta} = \frac{-2qJ\cdot[\rho - q\beta]+q\cdot[-c + q(1-2\beta)J]}{[\rho - q\beta]^2}$$

$$\implies \frac {\partial V(\beta)}{\partial \beta} = \frac{-2qJ\rho -cq + q^2J}{[\rho - q\beta]^2} = \frac{q}{[\rho - q\beta]^2}\cdot (-2J\rho -c + qJ)$$

$$\implies {\rm sign}\frac {\partial V(\beta)}{\partial \beta} = {\rm sign}\big\{(q-2\rho)J-c\big\} \tag {1}$$

So $V(\beta)$ is monotonic in $\beta$. Therefore, depending on the values taken by $J,\rho, c, q$, it will be optimal to set $\beta$ equal to either $0$ or $1$.  

Assume now that $\partial V(\beta)/\partial \beta &gt;0$. This requires that

$$(q-2\rho)J-c &gt; 0 \implies q &gt; \frac {c}{J} + 2\rho \implies q &gt; \rho \tag {2}$$

Then $\beta^* =1$ and

$$V^*|_{\beta =1} = \frac{-c - qJ}{\rho - q} = \frac{c + qJ}{q-\rho } &gt; 0$$

given $(2)$.  

In turn, assume that $\partial V(\beta)/\partial \beta &lt;0$. This requires that

$$(q-2\rho)J-c &lt; 0 \implies q &lt; \frac {c}{J} + 2\rho  \tag {3}$$

Then $\beta^* =0$ and

$$V^*|_{\beta =0} = \frac{-c + qJ}{\rho} $$

To have $V^*|_{\beta =0} &gt;0 $ we need

$$\frac {c}{J} &lt; q &lt; \frac {c}{J} + 2\rho \tag {4}$$

to respect also $(3)$. So here a necessary restriction on the parameters is imposed, in order to obtain a positive value for $V$.  

Combining, if we impose the restrictions

$$q &gt; \rho, \;\;\; q &gt; c/J$$

we allow for either $\beta =1$ or $\beta =0$ (it will depend on the exact parameter values), while at the same time guaranteeing that $V&gt;0$ in all cases.
This is not unusual in economic models -i.e. the need to restrict the parameters in such a way so as for the model to provide solutions that are economically meaningful. And we need $V$ to be positive, because otherwise the vacancy would not exist in the first place, as I understand the set up.