Used to think rodents in the coop were no big deal till I found one dead in the waterer last fall. Freaked me out enough to seal every crack I could find and switch to metal feeders. It's a legit disease risk so I'd toss that egg and give everything a deep bleach scrub just to be safe.
Metal feeders are a game changer. I learned that lesson the hard way too. Had a rat gnaw through a plastic one last summer. Did you find the dead mouse in the waterer before or after the birds started acting sick? That's the part that scares me most. Leptospirosis and salmonella can hit hard and fast with no warning. Bleach scrub is smart but I'd also dump that waterer and boil it if it's plastic.