There are tendrils, but the do not seem to be doing much.
I always cut the tendrils off and tie in my leaders.
It is a common practice to remove the tendrils on sweet peas so my methods are influenced by this.
How big a squash can a climber support?
Not sure really as I never weigh anything I grow.
I do know they can hold an average size butternut squash of around 6" diameter (measured at the bulbous end) without any other additional support.
I also find the 'tying in' helps with the weight situation, i.e. I always tie in where a fruit has formed, I do this with all my cordon grown stuff e.g. tomatoes & cucumbers.