Mostly at LAX but also had it happened at SFO when I flew in from RNO connecting to BR17 on the way to NRT. The trick is to be in which ever lounge EVA uses. At LAX that's the *A lounge in TBIT, at SFO it was the lounge that's now the Polaris Lounge, I think that they're at the lounge by G100,
According to the EVA website they use AF/KLM Lounge so that's where the EVA staff will be sometime around 2030. Since you're eligible for lounge access you should be able to get into the Polaris Lounge (at worst the United Club) before you head over to T5. My experience with the UA Lounge Dragons at ORD has generally been good, but I had to get a supervisor once when I didn't have a BP from OZ yet and there was an issue about whether I had access as an OZ *G, connecting to a *G flight (yes I got in, no where near the hassle at EWR when I've needed a manager a couple of times).. Just make sure you have all of the vouchers and a copy of your itinerary.
Hope you enjoy the trip!