By replacing $\phi$ by $-\phi$ you can set $\alpha$ to be positive (or negative, if you wish). By replacing $\phi$ by $\lambda \phi$ you can rescale away $\alpha$. So you can set $\alpha$ to be either $+1$ or $-1$ as you wish. 

Suppose $\phi$ is a $C^3$ solution on $\mathbb{R}^3\times \mathbb{R}$. Let $v = e^\phi - 1$; you have that $v$ is also $C^3$. Observe that

$$ \Box v = e^\phi \Box \phi + e^\phi (- \partial^2_t \phi  + |\nabla\phi|^2) = (1+v)(- \partial^2_t \phi) $$

Suppose first that $\phi$ is globally bounded: then $(1+\phi)$ is bounded below globally by some $\beta > 0$. Then the condition for Theorem 2 in John's 1981 paper is satisfied and the result follows. 

I think this can be upgraded to the case where the global bound is removed, but I need to check some details to make sure; I will update the answer if I find out either way.