The three Bug Crusher achievements are cumulative both through the Gaea campaign and the strategy game. To destroy a Walrus from direct control, you must be controlling either a Manta or Walrus of your own and shoot the final shot that destroys a Walrus controlled by the enemy AI. Any Walruses that are destroyed by your carrier, turrets, or vehicles that you are not directly controlling do not count for these achievements.
I would suggest completing the campaign first as you will need to destroy a lot of them to finish that. In a strategy game, invade an enemy island, starting by destroying any anti-vehicle turrets. When you are being approached by multiple enemy Walruses, save the game, destroy them, then reload the save. Rinse and repeat until you have 1000 kills, at which point the achievement will unlock.
If you choose to do this in the campaign, the first point at which you face enemy Walruses and can destroy them is after commandeering your carrier and attacking the second island, Arachnid. Note however that this stacks with the three "Air Strike" achievements, and as such you should not worry about these directly. Instead, focus on unlocking "Air Strike MK.III" and these three will unlock along the way.