How can I store leftover puff pastry or make it more durable?
No, you probably don't want to just squish leftover puff pastry dough together to save it.  While it would be perfectly safe, it would ruin the layering that gives the rise and flakiness that is the reason for using puff pastry.

If you have a large enough section still intact, you could put in the refrigerator for a day or two, to use later.

Otherwise, a typical thing to do would be to prepare sweet or savory snacks with the surplus pieces.   Cut it into convenient shapes.  If you have large enough pieces, sticks would be good, even twisting them to give them an interesting texture.

Then, brush the pieces with butter (or oil), and sprinkle with cinnamon and sugar; or grated Parmesan cheese; or rosemary and salt; or some other complimentary flavor.

Bake the pieces up until they are puffed and golden brown and they are very nice snacks or treats.