Nope. You need to have the OS drive inside of the system. For what it's worth though, I've been doing Windows Vista to 7 upgrades at work, having avoided doing "upgrade" installs my entire technician life, and I have to say, the Windows 7 upgrade process is very painless. I've got users with 500+ GB of sensitive data they can't lose and it does a fantastic job of keeping everything intact. You can also recoup some used space on the system by doing the upgrade. Just make sure you have at least 18GB of disk space free to do the upgrade, and you'll be fine. It's not any slower than doing a full install either - the upgrade install systems are doing great compared to the few that we've built with Windows 7 installed first.
The reason you can't use an external drive is to do with how the boot records are stored and how they access drives. I've also tried doing it myself, and it just doesn't work - you might get it to install ok, but the next time you disconnect the drive and try to run the system again after plugging it in, it doesn't work. Save yourself the hassle.