While not 1000% true, it is EXTREMELY rare for a bird to be able to transport zebra mussels from one body of water to another based on the training I attended. Frist, adult mussels need a solid substrate to attach to. Second, the immature mussels (veligers) are found in water and though they could be on a birds feathers or feet they likely don't survive most trips. And even after they get there must survive and reproduce.
Humans are the number one cause of zebra mussel transport from one body of water to the next. Any water left in a boat can have veligers in it and transfer to the next body of water. Adults attached to the boat are the other way. Fisherman with livewells are a big concern (those that don't completely drain) or water that is sitting in hoses. As others have said, ski and wakeboard boats with ballast tanks are another huge source of possible transport. Anything that can hold water is suspect.
Only way to kill them is a decontamination wash of super hot water. WY has one at each check station and at several major lakes along with multiple movable units. If there is even a question, you get decontaminated. Not that painful of a process with an average fishing boat.
Each state should be doing this IMO.