Are nafcillin and ceftriaxone eliminated via biliary secretion? Do you have to have a healthy gallbladder?

Yes and no. Nafcillin is excreted through the liver through biliary secretion. However, ceftrianxone is excreted mostly through the urine, and only partly through the liver/bile. However, with either one, you don't need the gallbladder for secretion. The gallbladder is only for storage of bile between meals. Even without it, you can excrete contents through biliary excretion normally.