Thanks to everyone who replied. I've downloaded Quick Settings and Android Assistant. Turning off the system settings with Quick Settings silences the shutdown sound, but not the startup. Android Assistant even has a setting which claims to silence startup sounds, but it doesn't work. The only thing that works is to mute from the home screen, which is what I want to avoid.
Somehow I silenced the original "Droid" sound on startup on my OG Droid, but I can't remember how. This is not a big deal, but it just bugs me. I'm not willing to root a two-day old phone, but thanks for that suggestion. It may be the only way. And why on earth does the Bionic (like my OG Droid) have to vibrate on startup and shutdown? Makes no sense to me.