You can offer straw from a feed store or coconut fiber than can be purchased at a pet store.

You can also purchase Coir/coconut fiber at a garden nursery. Some of them sell it by the foot.

dryer lint, chopped up bits of yarn or string.

Edited to add: if you have a dog or cat, collect their fur when you brush them.

Had a squirrel build a nest right where I could watch the progression and she used a lawn chair with cushion filled with polyester type cotton. That squirrel pulled out the fluffy cotton and used a lot of it to pad the nest. Got any old pillows or cushions?

Or a few cosmetic cotton balls.

I don't think that the squirrel will leave you alone and stop gnawing on your outdoor rug in this way. The squirrel likely drags the pile from your rug to make a nest for itself.

I think that the squirrel will understand that you can be manipulated in this way and will continue it. These animals are smarter than they might seem at first glance. On the website https://campingfunzone.com/2021/08/20/squirrels-3/, there is a whole huge article about this. I think this article will help you understand the psychology of squirrels and help you understand how to deal with them.

I must say right away, it was very difficult for me to get rid of squirrels. It was a whole war.

"...more than 20 squirrels who come to eat and play with my children."

Your children need Rabies Vaccinations - at least!

INCREDIBLE!