Bigels (BGs) are innovative composite systems that integrate oleogel and hydrogel structures, and are gaining increasing attention for their unique textural and functional properties in food applications. This study evaluated the rheological and mechanical properties of egg white-based bigels incorporating candelilla wax (CW) as an oleogelator. The results indicate that different egg white protein (EWP) (5–10%) concentrations and hydrogel-to-oleogel ratios (20:80 to 80:20) significantly influenced the structural and functional properties of the bigels. Compression testing revealed no significant differences in strength across the tested range; however, higher EWP concentrations enhanced the stability of the BGs. Furthermore, increased candelilla wax oleogel (CWO) content (60%) markedly improved emulsion stability, resulting in superior strength, as confirmed by dynamic light scattering. Rheological studies demonstrated shear-thinning behavior, particularly at higher hydrogel content related to the oleogel (W/O), which exhibited the highest yield stress. Microstructural investigations confirmed the presence of a continuous oleogel phase within the bigels (W/O) and revealed the formation of a complex structure. These findings suggest that a reduced hydrogel-to-oleogel ratio can be utilized across various food systems, opening new possibilities for creating customized food structures with desirable textural and functional attributes.