CI is far superior to BR in BIz.
The plane you will fly on from LAX is almost certain to be very new 747, and row 10 will give you massive room.You can pre book this when you book your ticket. Downstairs is good also with the window seats having the storage bins as well as overhead space.
Service is good on both but again I find CI have overtaken BR ( I fly both a lot ) You can pre-order your meals on line once you have your flight reference.
The plane they have between TPE and BKK is either 747 or Airbus. The airbus has almost lie flat seats and is better than the 747.
The lounge in TPE is amongst the best in the world in T1. You can get between T1 and T2 very easy with the monorail and it takes about 2 minutes.
Biz on EVA is pretty cramped and the product is out of date. If you get the new 2 class config this is actually very good as I found out in December.