Yes they do push other minions, but not off the board or through a blocking obstacle.
From field guide:
Move Commands move the Mech to another space, while facing the same direction it started. All Moves except for Omnistomp will only move the Mech forward. Omnistomp can move the Mech forward or to either side, but not backward. If the Mech moves onto a Minion, the Minion is stomped and killed. If the Mech moves onto another Mech or the Bomb, the Mech or Bomb is pushed along in front of the moving Mech.
You must move unless you are blocked by something impassable.
Also from watch it played:https://youtu.be/FLtVB6zkvcs?t=8m3s