The way to setup the regression depends on what do you want to predict. Once you formulate exactly what do you want to predict, you should set up your regression in exactly same way.
Daily regression of returns of JPYStock ~ SPX can be done in several ways, and you should consider these differences:
- holidays as you mention
- End of trading day time (i.e. one will be ahead of other, thus there is information leak)
- currencies are different, i.e. you might want to include USDJPY spot FX in your regression
Even if you disregard these differences, and just focus on daily regression (which is basically just estimating Pearson correlation and vols,ignoring FX moves, and accepting that there is information leak), then these holidays differences do not matter if during the holidays period the returns are not abnormal. Estimating correlation over few years, and adding some dozen of non-abnormal returns will not change this correlation significantly. (by adding i mean substituting the returns by 0 if not available on holiday date)
as noob2 mention you might want to calculate this correlation over longer periods, such as weekly, monthly etc. This will change your correlation more than effect of including/excluding same holidays.