Blue Hubbard Squash
When you say "when the vine starts dying off," what does that mean exactly? The leaves on my vine are dying, starting at the base of the vine and progressively working their way to the end of the vine.
I assume your wondering about harvesting your squash.
The vine will start to turn brown and lose it's green color. The leaves start to turn slightly papery as the vines and plant die off.
The last paragraph of this article will help you.
The stem of the squash is a good indicator of when the squash is ready to remove from the vine.