In Zen it is taught to keep ones eyes focused downwards. Would it be different to hold eyes horizontally directly forward? To me this seems to silence the mind more easily. In some schools of yoga eyes are held looking forward and up as if looking a star in the sky. What's the difference?

It depends on the person as to whether they are tired or not and how their strength of concentration is. Raising the focus of your eyes is helpful, when you are tired. Yet it is not possible to enter samadhi when looking straight forward. We can use objects of focus to strengthen our concentration, looking at a stone for example, but once we enter deeper concentration, this dualistic experience falls away and the stone is of no matter anymore. This is one technique but not the way to become one with the universe. You can stare at something and feel like you are concentrating, but it turns into intoxication. For true samadhi and concentration, all dualism, all division between outside and inside have to fall away.