A series of hydrogels containing sodium alginate at different concentrations (2%, 3%, and 4%) and egg white were prepared through ionic cross-linking with calcium chloride (CaCl) to obtain composite dressing materials. ZnO nanoparticles coated with eucalyptus or lavender essential oil were introduced into the hydrogel matrix to enhance antibacterial properties. The resulting hydrogels were freeze-dried to enhance mechanical properties, increase the porosity of the dressing, and facilitate further evaluations. A variety of analytical methods, including scanning electron microscopy (SEM), X-ray dispersive spectroscopy (EDS), and Fourier transform infrared spectroscopy (FT-IR) were employed to characterize the composites. The developed composites exhibited high porosity and a swelling degree exceeding 200% after 3 days. Additionally, water absorption capacity increased with higher alginate concentrations in the samples. Furthermore, they demonstrated significant antibiofilm activity against , , and , with the samples containing 4% alginate showing the best results.