First: I'm assuming you want a cuboid not a cube. A cube has the same height, width, and depth - always. A cuboid is a rectangular prism.
Second: XNA doesn't let you dynamically create the "Model" class (Or, at least, not easily). So you have two choices here:
Option A: (The easy fix) Create a perfect 1x1x1 cube model, use that for everything, but, when drawing, scale it with a Vector3 set to your height / width / depth choice.
Option B: (The pointlessly hard way) Forget everything you know about 3D models in XNA (Specifically the sections about the Model and ModelMesh classes), and build VertexPositionNormalTexture[] VertexArray and int[] IndexArray and render it using a BasicEffect object and device.DrawUserIndexedPrimitives(PrimitiveType.TriangleList, ... )
If you choose option B, note that a proper cuboid has 8 Vertexes, and 12 triangles (36 indices)