On another forum this same question was asked and a few of the responses were that the reason for them closing their eyes is because they are ignoring you.

This could be because she feels safe when she closes her eyes or because she doesnt want to bed bothered. A lady posted some pics of her iguana closing one of his eyes (eye that was closest to the lady) but the other eye (eye farthest away on the other side of his head) would be open.

I tend to think its more of a feeling safe thing. When she closes her eyes she isnt scared because she cant see any danger.

It takes time for her to get used to you. This is my first iguana but I have had water dragons and other reptiles and Im sure its the same with an iguana as it is with them. Just keep handling her and being around her to show that your not going to harm her and she will get used to you. Soon you will be able to lay on the couch and watch tv together haha....

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