I would suggest that you use the bootstrap. Briefly, you would take the error terms from each of your two regression models and create 95% bootstrap confidence intervals for the two variances for each of the two sets of the error terms. An easy way to address your query would be to look to see if the confidence intervals overlap (or not).
Thank you Dragan. Suppose I found that these intervals do not overlap when using the 95% confidence interval, how would my conclusion incorporate this value of 95%? Could I state that there is 5% risk the variances are identical?