Delivery of chemotherapy drugs specifically to cancer cells raises local drug doses in tumors and therefore kills more cancer cells while reducing side effects in other tissues, thereby improving oncological and quality of life outcomes. Cubosomes, liquid crystalline lipid nanoparticles, are potential vehicles for delivery of chemotherapy drugs, presenting the advantages of biocompatibility, stable encapsulation, and high drug loading of hydrophobic or hydrophilic drugs. However, active targeting of drug-loaded cubosomes to cancer cells, as opposed to passive accumulation, remains relatively underexplored. We formulated and characterized cubosomes loaded with potential cancer drug copper acetylacetonate and functionalized their surfaces using click chemistry coupling with hyaluronic acid (HA), the ligand for the cell surface receptor CD44. CD44 is overexpressed in many cancer types including breast and colorectal. HA-tagged, copper-acetylacetonate-loaded cubosomes have an average hydrodynamic diameter of 152 nm, with an internal nanostructure based on the space group Im3m. These cubosomes were efficiently taken up by two CD44-expressing cancer cell lines (MDA-MB-231 and HT29, representing breast and colon cancer) but not by two CD44-negative cell lines (MCF-7 breast cancer and HEK-293 kidney cells). HA-tagged cubosomes caused significantly more cell death than untargeted cubosomes in the CD44-positive cells, demonstrating the value of the targeting. CD44-negative cells were equally relatively resistant to both, demonstrating the specificity of the targeting. Cell death was characterized as apoptotic. Specific targeting and cell death were evident in both 2D culture and 3D spheroids. We conclude that HA-tagged, copper-acetylacetonate-loaded cubosomes show great potential as an effective therapeutic for selective targeting of CD44-expressing tumors.