Green Amoebae are large and mutated, presumably single-celled organisms like the Bacterium that appeared in Parasite Eve. Green Amoebae can be difficult to see because their dormant state is extremely small and semi-transparent, making it is easy to ambush Aya.
Green Amoebae attacks are two-fold. They crawl over to Aya to make physical contact so they can leech MP off her and inflict SILENCE, preventing Aya from casting spells. Afterwards, they charge up a spell similar to Pyrokinesis and launch it at her. Given how slow they are to turn, this attack is fairly simple to dodge and can even hit other NMCs on the field for damage (which includes setting Maggots on fire).
POISON and BURST attacks do a special one-hit kill damage with a range of 280-407 damage across all difficulties. Necrosis at all levels is the cheapest way to achieve this (though at level 1, close-range is required). 44 Maeda SP and Grenades will also work, but given they are a far more expensive ammunition, they are not recommended.
All melee attacks will harm an amoeba, though the Tonfa Baton is granted a special boost of 500% bonus damage and requires the least amount of target leading. A couple swings should be enough to take an amoeba out. Given the amoeba's attacks only inflict MP damage, combined with the generous amount of MP an amoeba provides at the end of battle makes melee attacks a viable option.
Green Amoebae are unaffected by status effects, but there is one way to neutralize them. Despite appearances, Green Amoebae are affected by 0 dmg. attacks. If they take 20 damage within a certain amount of time (Approx. 2.13 seconds before the counter resets), they will be temporarily stunned. This lasts for 12 seconds, making contact with it does not harm Aya and can be interrupted if they take a hit from something that does regular or special damage.
Amoebae appear to be sensitive to any use of PE energy. Weapon attacks will only alert the one being hit, however, casting a PE spell will alert ALL the amoebae in the room, regardless of distance.
Amoebas are extremely vulnerable to burning Maggots. Contact with one will do rapid burning damage and likely kill it in an instant.