During extraction, transport, and logistics activities of crude oil, it is necessary to use methods to prevent operational problems caused by waxes precipitation, which can significantly increase the oil viscosity or even stop its flow. To overcome this problem polymeric additives can be used. Recently, research has indicated the possible advantages of using nanomaterials for this purpose, and polymer:montmorillonite has been explored. In this work, we produced nanocomposites of EVA and a clay mineral: montmorillonite, palygorskite or halloysite. Clay minerals were organically modified by two different methods. Nanocomposites were produced by the melt and solution methods, with different polymer:clay contents, using two drying procedures. The materials were characterized by different techniques. Performance tests of the additives were carried out in waxy model systems (varying the wax chain lengths and its concentration) through pour point analysis, microcalorimetry, rheology and optical microscopy. All nanocomposites were able to reduce the pour point of the system containing waxes with high molecular weight (melting point of 56–58 °C), while pure EVA did not have any effect. The nanocomposites, with 95:5 and 90:10 wt/wt%, were able to reduce the pour point from 21 to <−24 °C for systems containing 5 wt/v% waxes. The efficiencies of these nanocomposites gradually decreased when wax concentration was increased from 5 to 5.5 and 6 wt/v%. Nanocomposites with palygorskite were the most efficient in reducing the pour point and viscosity. For systems containing 5.5 and 6.0 wt/v% of waxes, the pour points were reduced from 21 to <−24 °C and 22 to 10.5 °C, respectively. This behavior was due to the different structures of clay minerals, which interacted differently with EVA and, consequently, with the waxes in the systems. The nanocomposites presented the following efficiency order: EVA:palygorskite > EVA: halloysite > EVA:montmorillonite. Although the nanocomposites did not modify the wax appearance temperature (WAT) or influenced the quantity of precipitated wax below the WAT, they acted on the crystallization process, reducing the crystals size. The efficiency of the nanomaterial was influenced by its composition, the organic modification method of the clay mineral, and the method of preparing and drying the nanocomposite.