Can Eye Bags Be Caused by a Muscle?

The bulge you're seeing is likely just fat. Muscle bulges in the lower eyelid are usually seen only when you smile. If you like what you see when you cover the trough (lower eyelid groove), then you could consider an Injectable Filler treatment to fill the groove.
You should consult a board-certified dermatologist or facial plastic surgeon experienced in the delicate art of filling lower eyelid grooves. You should see many photos before proceeding.
I've attached a link to my Injectable Filler photos for your perusal.
I hope this is helpful for you.

The pooching you see in the lower eyelid bags is the fat that normally sits under the eyeball being pushed out of the socket as it is no longer kept inside by a looser SEPTUM, eyelid MUSCLE (Orbicularis Oculi) and on occasion EXTRA lower lid SKIN. As our cheeks begin to droop with aging, we lose the look of youth where it is hard to see a wide lower lid - instead the cheek seems to extend to the lashes. As the descent continues, support to the lower lid decreases and intraorbital fat is pulled out ballooning the lower lid. The dropping cheek unveils the lower ledge / rim of the orbit which DIRECTLY underlies the TEAR TROUGH (feel for yourself).
"When I close my eyes, the bags disappear and the underneath looks good" - because you contract the Orbicularis muscle, which then pushes the fat back into the orbit. As soon as it relaxes, fat pushes right out. I would bet you that if you lay on your back and looked at your eyes with a mirror the bags would be gone too . This time gravity, not muscle contracture, put the fat back in the sockets.
The Tear trough deformity can be treated with Restylane. The eyelid puffiness would require a more in depth examination but could be corrected with a lower Lid Blepharoplasty.
Peter A Aldea, MD

When you squint, the sheet of muscle that surrounds the eye contracts and pushes the fat in the bags back toward the eye and smooth out the area. Sounds like some of the fat needs to be removed and maybe placed in the tear troughs.