The evacuation process during the Egg Hunt mission is very similar to that found in mining missions. After completing main and the optional side objectives, players can initiate the evacuation process by pressing the appropriate button on the M.U.L.E. robot.
Players have 5 minutes until evacuation - if they don't do that within the time limit, then the mission will fail. It is worth noting here that the escape vehicle in Egg Hunt missions usually lands slightly closer than in the mining missions - you don't have to run through the entire cave.