As a general rule, winter squash can stay on the vine for as long as the vine is alive and the squash's stem is not browned.
So, when the stem is semi wooded, you can be sure that the fruit is not getting anything from the vine. THen you can pick them.
But like most fruits, they can continue internally to ripen and get sweeter, at the expense of loosing moisture.

another test of ripeness is finger nail test. If you cannot easily scratch it deeply, then it is ripe.

With butternuts I want the green stripes to fade until you can hardly see them, and the color of the rind should change from pinkish beige to a nutty brown. The vines are usually not completely dead. Curing the squash in a warm sunny spot for a couple of weeks will "finish" butternuts that are a shade short of perfect.

Planatus has it just about the way I do it. The squash start out green, then turn light tan, with longitudinal green stripes. Those stripes get thinner and thinner, and finally they disappear, at which point the squash should be completely tan/beige. I often let them sit even as the vines die, and pluck them up when the vines are no more. Don't forget to dip the harvested squash in a dilute bleach solution to sterilize the skin. That will make them last longer. I believe that if the vines are dead and there are still thin green stripes, those stripes will eventually go away by themselves.