Where does this equation in the electrophysiology literature form come from?
They chose that equation mainly because of numerical simplicity. It can fit the rates $\alpha$ and  $\beta$ for all the ion channel particles in the model. 

The theoretical foundation for the equation, as in the H&amp;H model, are loose. It resemble the equation derived for the movement of a charged particle in a constant field. But it is not an equation derived from the ion channels biophysics. We can't blame them, considering that at that time (1952) they don't even know for sure that ion channels where proteins.

In page 183:
 In order to simplify the reporting of the actual values used in this model, we have expressed the alphas and betas entirely in terms of a generalized function, with eight defining coefficients. Table 1C gives the equation, and the defining coefficients for al rate constants.

For the generalized function note that in every case at least some of the coefficients are zero: this generalized formula is just a way to collect both possible forms for the H&amp;H rate constants.

I have read papers using Eyring equation for a theoretical formulation of transition rates, based on thermodynamics.