Why is my flying squirrel's coat greasy and losing hair? Is it diet?

This is most likely diet related. I don't know what you are feeding her but I would suggest you review what you are feeding and make sure it is a correctly balanced diet. Almost every medical issue we see with these guys is due to improper diet. A good flying squirrel diet consists of a combination of fresh nuts and fruits.

* Nuts may include peanuts, pecans or walnuts.

* At least 40% of their diet should be fresh fruit. Fresh fruits may include apples, grapes, oranges, peaches, pears or plums.

* Other foods that may be included in their diet are carrots, mealworms and sunflower seeds.

* A commercial flying squirrel food made by Brisky should also be used.

( https://brisky.com/index.php?_route_=flying-squirrel-diet )

* Provide water in an open container, not a water bottle.

* A spool of rabbit salt should be kept available at all times.

Hope this helps. If you have more questions you can always opt for a private consultation where we can chat back and forth. That way you can ask as many questions as you'd like. Best wishes.