Is there a way to require a number of user points in order to sign up for a node?

Yes. See signup.api.php that ships with the module. There's hook_signup_sign_up(). However, that won't let you actually prevent a signup if a user doesn't have enough points. But, you could at least deduct points whenever someone signs up (but nothing would stop them from signing up if they had 0 or negative points).

To prevent signups to users without enough points, hook_form_alter() is probably all you need -- just add another validate callback to signup_form() that checks if the user doesn't have enough points, and if not, throw a validation error. This will only handle users signing up themselves, you'd have to alter some other forms to prevent admins from signing up users without enough points if you want that behavior, since there's no global hook_signup_validate_signup() or something.

Grab the latest copy from DRUPAL-6--1 branch and/or the 6.x-1.x-dev release.