I put my best bikes on my roof rack and drove across the country with the bikes on top of the car.
That said, I packed some bikes in bike boxes got from the local bike shop. To well protect the bikes, you wrap every frame tube in bubble wrap. You can also surround the bike with other softer stuff (towels, clothes put in plastic bags, etc). Also, you need to be on site when they are packing the truck and tell them that they are bicycles and need to stand upright and not be underneath something extraordinarily heavy.
I couldn’t fit a bike plus wheels inside the bike store box. I don’t know how the bike manufacturers solve that puzzle. I got an extra bike box and put wheels in it with other soft stuff.
I would not trust a mover to pack a bike. Some of those guys are really good at packing trucks but most of the helpers are low wage labor.
If you really want to be extra safe, buy a hard sided bike travel case.
Moving insurance sucks so everything you pack should be able to survive at least a 6 foot drop in it’s box. It should also be able to be the box on the bottom of a 200-300 pound stack of boxes. If you pack like that you’ll be fine.