I just flew on EVA from San Francisco to Bangkok in July with a MONO M80 gig bag and I was allowed to carry it on. They let me store it in a wardrobe closet used for the flight crew's stuff. And that was on two separate flights, San Francisco-Taipei and Taipei-Bangkok.
I talked to the check-in counter and very politely asked if I could carry on my bass case. At first they said "No", but I (falsely) explained that it was a gift from my grandfather (I bought it on eBay) and that I was afraid that it would be damaged if I checked it in (this was true). Then they told me to ask the flight crew when it was time to board since they have the final say. I gave my sob story to them and they said "Yes". They even let me board with 1st class so I wouldn't be in other passengers way while I stowed the bass. There was another guy with an acoustic guitar in a gig bag and we both put our guitars in the same closet.
Bottom line: be polite, smile, and mention grandparents. I found the EVA staff to be very accommodating. Like someone else said, you'll probably have better luck with a gig bag vs. hard case.
"Literally everyone is a musician if they want to be. Good for culture, bad for employment."