When someone shuteyes, they close their eyes to go to sleep.
Shuteyes are usually good because it means someone is resting and getting ready for a new day.
Examples:
She shuteyes before the bedtime story.
I shuteyes when it gets dark outside.
He shuteyes in his cozy bed.
The baby shuteyes in the crib.
I shuteyes for a long time when I'm very tired.
The cat shuteyes in the sunbeam.