The mosquitos will be the most difficult I think.
The only thing I can think of that would allow you to keep a squirrel outside would be to dose her once a month with Revolution. It kills ticks and fleas plus a number of other parasites and is well tolerated. It is prescription only but can be purchased here:
https://www.canadavetexpress.com/rev...k-pack-45.aspx
You would NOT be using a whole vial to dose each month so it will last a lot longer. Depending on her size it would be only 1-3 drops on the back of her neck.
The only thing I can think of for the mosquitos would be to get mosquito screening (for windows) and install it on the outside of the cage where her nails don't get to it so much.