I did some experimenting and figured out my power issue. I have been using the Nexus 7 charger with a longer USB extension cord since the 3 foot cord that came with it just wasn't long enough most of the time. However, I installed the Battery Monitor Widget app and discovered that while connected to the charger with the 6 foot extension cord, my Nexus pulls between 300 and 500 mA from the charger (and uses about 1000-1200 mA while running BOINC), but when it is connected to the charger with only the short cord that came with it, the Nexus pulls between 1100 and 1400 mA (enough to counteract the drain caused by running BOINC). I didn't expect that a longer cord would have any effect, but apparently either the charger or the Nexus 7 (or both) don't like it for some reason.
The short cord also charges the tablet a lot faster while it is idle, so I guess I need to remember only to use the USB extension when I actually need it, and just plug in the short cable any time I'm running BOINC or when I want the Nexus to charge faster.
Unfortunately, the Einstein@home tasks started out running quickly (about 5% in the first 20 minutes) they have slowed down considerably since then and look like they will take about 20 hours to complete, just as the Asteroids@home tasks did.