Hello, I'm Dr. Deb and will do my best to help with this concern about Dax.
It's certainly possible that her eye is just irritated but if she's keeping it closed, then this usually signifies pain or discomfort as might be seen from a scratch or injury to the cornea.
If the discharge is green or yellow, then this may mean that a secondary infection if present.
I do have several acceptable otc options which I'll list below and which can be safely used in her eye; however, if you don't see improvement in a day or so and/or the eye looks worse, then a vet visit may be wise.
The Systane products which are listed can be used but the ones with vasoconstrictors, such as naphazoline should be avoided. These products are labeled as "taking the red out" or for "red eye".
Tears Naturale Nighttime
Systane Ultralubricant sterile eye drops
Advanced Eye Relief Nighttime
Akwa Tears ointment
I recommend treating 3 times a day.
I hope this helps, Deb