No!!! Your gerbils will not catch your cold!
Scientists have been trying to infect various species with the rhinovirus (the virus responsible for the common cold) for years to help them understand its mechanisms and it has been very unsuccessful!
“PROGRESS toward the chemotherapy or prophylaxis of human, rhinovirus infections would benefit from studies in suitable experimental animals, but our attempts to produce rhinovirus infection in several small laboratory animals including leopard frogs, guinea-pigs, gerbils, marmosets, and newborn, as well as adult, mice, ferrets and hamsters, have been unsuccessful.” Experimental Infection of Gibbons with Rhinovirus -CARL A. PINTO & RICHARD F. HAFF
However they have been having limited success with gibbons, so if you have any pet gibbons around the house don’t splutter on them. 
As for the flu, it depends which strain you have. Influenza A is most commonly hosted by wild birds and only H7N7 has any real zoonotic potential as it has infected horses and lab mice. Influenza B is almost exclusively a human virus and influenza C virus apparantly only affects humans, dogs and pigs, but is much less common.
If you do have the flu however, I’d imagine handling and playing with the gerbils will not be a priority on your daily to-do list, especially if you get that wonderful symptom that involves your latest meal to exit your body in a hurry!
It’s not like a week or so of little to no interaction will cause the gerbils to become wild and aggressive, so if you have nobody else that can check their food & water for you, just throw in some cardboard and food and just top up the water every couple of days. Make sure you take care of yourself, the gerbils will survive 5-8 days of less interaction! 
If your gerbils are wheezing, clicking or looking generally unwell, a trip to your vet is always worthwhile, just don’t let yourself be fobbed off with a bottle of Baytril. If your vet insists on a course of antibiotics, insist on using Marbocyl or Trimethoprim sulfa.
Baytril has a weakened impact on bacteria nowadays (thanks to all of the idiots breeding on gerbils that have been treated with it) and many gerbils have bacteria that is now partially immune to it, not to mention that Baytril will lower the seizure threshold for your gerbil (meaning potentially more seizures if the gerbil is already prone). Your vet will forget that tidbit of course, so feel free to remind them.
Stay safe and more importantly, keep your germs to yourselves!
