Whatever you do, please do NOT stow it in the overhead bin. UA doesn't allow this because they had someone permanent nerve damage when a luggage cart came down on someone's head. You would be careful but the idiot in a hurry two rows behind you wont. He'll grab his stuff and down it will come on some poor little old lady. Best avoided...

To reduce damage to any stroller;
-"Strip" it. Take off anything you don't need in the airport, including the sunshade (depending on the model).
-Attach the straps
-Double secure it with a bungee cord
-Bag it. Don't feel obligated to buy any special bag. One poster here uses a laundry bag. It's just to keep it clean.

It's a pretty compact stroller so be nice and maybe they can slip it in a closet. Just ask the F/A's quickly when you board. Don't bother getting any "permission" ahead of time because it will just depend on how much stuff is in the closet.