Rather, he worries pet owners might abandon their animals at the first sign of a sniffle.
"I think that's what's going to wig people out," Johnson said. "I don't want to see the shelters filled with cats and dogs tomorrow."
Veterinarians have long heard of the flu jumping from animals to humans, and some cases of pets to humans. But it's uncommon for a flu virus to jump from a human to a cat or dog.
"The H1N1 virus obviously has the potential to jump from animals to humans -- most people believe that that's where it originated -- and it's already been proven to make the jump from humans to animals," said Dr. Steven L. Rowell, director of the Foster Hospital for Small Animals at Tufts University.
But ferrets and pigs, rather than dogs and cats, seem to be the animals most at risk for catching a human flu. The new H1N1 strain is no exception to that trend.
The United States Department of Agriculture has documented one case of a ferret in Oregon contracting the flu from an infected owner and several pigs contracting the H1N1 virus from humans.
Although ferret owners are experienced in dealing with human-to-ferret flu, dog and cat owners may not be, and veterinarians have some specific recommendations.
If an owner is worried that their cat or dog has the swine flu, veterinarians recommend that owners not rush to the local vet's office demanding an H1N1 test.
"If people are going to their vet with their cat or dog, there's not, to my knowledge, an easily accessible way for vets to test them for H1N1," Johnson said.
To protect the household from pet flu infections, veterinarians recommend the same precautions families should take when a person falls ill.
"Wash your hands before and after touching your pet, and avoid sneezing on your animal," Rowell said. "And if you suspect your pet has the flu, do not treat him or her with drugs yourself, bring them in to your veterinarian. Treating them with over-the-counter flu medicines may do much more harm than good."
The Associated Press contributed to this report.