If you're only using one texture, and want to change ALL the parts, the easiest way to do this is to set the MaterialPropertyBlock for the Renderer.
The code would look like this
void Start () {
MaterialPropertyBlock mpb = new MaterialPropertyBlock();
mpb.SetTexture("_MainTex", myTextureToUse);
GetComponent<MeshRenderer>().SetPropertyBlock(mpb);
}
More info here: https://github.com/pharan/spine-unity-docs/blob/master/Rendering.md#setting-material-properties-per-instance
Note that this will not work if your atlas requires more than one page.
Otherwise, you can always use Spine's skins feature.
If you want to use Mecanim state machines, you can use SkeletonAnimator "Instantiate (Mecanim)" instead of native SkeletonAnimation.
It creates dummy Unity AnimationClips and it will work with Mecanim and Animator components. We only recommend using this if you really want to use Mecanim and are familiar with its quirks. There are other people in the forums who use it.
Beyond that, we can't offer much support for unofficial code (Spine2Unity). This one in particular not only works very differently from the official runtime, it's also closed-source. So you'll have to contact that person who made it for support there. Sorry.