Yes, you can control Spine assets in Unity using code! There are several ways to do this.
Basic Animation Control
You can control which animations play and manipulate bones programmatically. The spine-unity runtime provides full access to your skeleton's bones and attachments.
Controlling Bones from Code
spine-unity allows bones to be controlled by Unity GameObjects. You can set up GameObjects to position specific bones like pupils, and then control those GameObjects with your scripts.
Getting Started
To learn how to control your Spine assets in Unity:
Check out the example scenes included with spine-unity, particularly the 5 Basic Platformer example at Spine Examples/Getting Started/5 Basic Platformer. This example demonstrates various ways to control Spine characters through code: http://esotericsoftware.com/spine-unity#5-Basic-Platformer
Look at the example scripts, especially Spineboy.cs, which shows how to control animations and interact with the skeleton: EsotericSoftware/spine-runtimesblob/master/spine-unity/Assets/examples/spineboy/Spineboy.cs
For more advanced control, the spine-unity runtime documentation provides comprehensive information on the API: http://esotericsoftware.com/spine-unity
The spine-unity runtime gives you the same level of control you have in the Spine editor, allowing you to manipulate bones, play animations on different tracks, and create dynamic, interactive characters!