I just got eggplants starting to grow, with some flowers too. But I'm not sure if I should put a cage around it, like a small tomato cage.
Support is good for eggplants as the fruit may become heavy and can even break the branches.
I have used tomato cages successfully, making sure the fruit are not trapped "inside" the cage, and ensuring there is good airflow through the plant. This is for pest control access, and so the plant does not rot, especially in hot, humid and/or rainy weather.
You can also just use a stake and tie the plant loosely to the stake to support the branches with the flowers (that will become the fruit)
I use tomato cages on my pepper plants as well.
Any support is good. Eggplants and Tomatos (among others) are nightshades and have very similar growing requirements.
When growing eggplants or Tomatoes commercially there is generally support by way of strings suspended from wires which can be reeled in as the plant grows. There are plastic clips at the bottom of the wire to support the plants.