As soon as you reach the first Shieldbug encounter, have one marine run past them while the rest disables them using the items mentioned above. The rusher should take care of the eggs located on the right side of the map while the rest of the team takes care of the eggs on the left side. Regroup at the final egg area, blow up the barrel, then run ahead and ignore the Mortarbug. Disable the first Shieldbug quickly to stop it from blocking your path, then lure the second Shieldbug out and disable it so you can run into the exit zone.
This mission may be easier to complete with only Faith and Wildcat, due to their Speed Bonus. Slower marines may have a problem bypassing the Shieldbugs in the middle and those blocking the exit. The squad's time is only as good as that of the last marine to make it to the exit.
This achievement can also be soloed using only Wildcat. Bring a Grenade Launcher, a case of Adrenaline, and a backup weapon to kill Drones. Use the same general strategy as above, but with liberal Adrenaline usage to get you through the eggs as cleanly as possible.