I haven't had the time to test this, but after looking through the sample's code I believe you'll need to do a few changes for it to work. Here's what I have in mind, although I'm not sure it will work as it is:
For starters, the way the particles are being billboarded at the moment doesn't allow for much variation. Each vertex basically carries the particle position and a 2D corner offset such as (-1,-1) or (1,1).
The GPU does all of the particle animation, applies the view and projection matrices to get it into clip space, and then adds the corner offset multiplied by the particle's size to the clip space XY position. Since this sum is done in clip space, the resulting quad will always be facing you. No billboard matrix anywhere.
You'll need to change your shader in order to apply the billboards in a more traditional way using a matrix. Using the method r2d2rigo mentioned (Matrix.CreateConstrainedBillboard) you'll need to calculate a billboard matrix for each particle and pass it to the shader when drawing.
Then you'll need to change the method in the shader that is currently applying the View and Projection matrices and leave that part out, because you still need to do a few other things before converting to clip space.
With the position still in world space, that's when you will need to apply the corner offset, along with the billboard matrix you created before to make it face the camera.
Finally just apply the View and Projection matrices as it was done originally in the sample.
Start from the billboard sample instead and adapt it to your particle needs. That might turn out to be easier.