The horse comes into physical contact with something everytime it enters an enemies square. Each time it does this it may move the enemy 2 squares back or trample them. There is no save for these effects.
You're talking about the effect of a bull rush, in the first part, pushing an enemy back. While this is a viable tactic, the horse may have a difficult time pushing back a larger opponent. You also state there is no save for this attack. It would not be a save anyway; it would be an opposed check (probably opposed Strength check; I’d have to double-check the Bull Rush rules). If the horse wins, the opponent is pushed; if the opponent wins, they don’t get pushed (the Bull Rush attempt may also provoke an attack of opportunity (AoO) – also not sure on that).
Also, you talk about the horse entering an opponents square; this would provoke an attack of opportunity from the opponent. This attack may be negated when you’re using the horse in conjunction with the Swarm of Toys, but that would be a special case, which should be documented separately.