Simply put, unfortunately you can't run two vastly different signals from a single port. In this case, you would need to run them on different frequencies, probably different resolutions, and even having them on extended mode (so not cloned). Furthermore, you would need to run both analog and digital signals within the same cable. Not possible. You might have to fiddle a little bit even if you run them on different ports (the frequency difference is the biggest concern, I think).
In any case, your only option would be to run the VGA monitor through the GPU's DVI port, which is currently your GPU's only port supporting analog signal transmission. If I understood correctly, this is the current setup?
As for 120/144Hz, your GPU's only options are mini-DisplayPort and DVI (assuming it supports DL-DVI). Unfortunately, the BenQ doesn't seem to have a DisplayPort of any kind on it, and the DVI would have to be preserved for the VGA. Therefore your only option would be to try a miniDP-to-DVI adapter (edit: link removed, not applicable), but I have no certainty whether that will work with 120/144Hz (edit: only active adapters might, but apparently not all of them will). But other than that, there aren't any other options I can think of with the current components.
(and this is assuming that the previously stated "mini port" is indeed mini-DisplayPort)
(also, I wouldn't be so sure that there's a DVI cable included with the monitor, so you'd better check before purchase -- and if not, then make sure you get a dual-link compatible cable)
(edit: judging from the user reviews, there is a dual-link DVI cable included with the BenQ monitor)
In summary:
Old monitor: GPU DVI --> monitor VGA (with the current cable)
New monitor: GPU mDP --> monitor DVI (with mDP-DVI adapter + DVI-DVI) (edit: will cost around £80, and no guarantees it would work)