You would need to use the Surfaces palette in D|S - View>Tabs>Surfaces. I strongly recommend selecting Show surface list from the palette's option menu (the triangle button at top-right).

In the surface list (on the left of the palette if you took my advice, a drop-down at the top if not) you should see an entry for the horse, with a list of surfaces underneath it. In the right-hand half of the palette there will be a set of controls - click each surface under the horse in turn and note what's listed in the long button in the Diffuse section (if there isn't a long button with a name, just the colour bar with a triangle button next to it, then there isn't currently a texture applied to that surface. I think you will find that every surface has the same texture - the Poser horse doesn't use multiple maps - so you should be able to select all of the surfaces (click on the first, shift-click the last), then click the buton with the current texture name (or the triangle button next to the colour bar), select Browse for image..., and load the texture you want to use. You may want to click the colour bar and select white as the base colour, if it isn't already (especially as Bryce doesn't let you apply both a solid colour and a texture).

Once done, assuming the horse looks ok, make sure you have it seelcted in the scene and go to File>Save as>Material preset...; that way you won't have to reapply the texture by hand next time you want to use it but just use the material preset instead.