A cookie is a cookie is a cookie.
As long as the code from asp or asp.net knew what to look for in that cookie, it shouldn't matter.
I'm not sure about the browser closing thing (I'm sure someone here does), but you could just set the lifetime of the cookie to be something like an hour or so, assuming you knew that most users wouldn't be hanging around that long (1/2 hour might even be doable).
Jack