Maybe. It depends on whether or not you have a CD/DVD of the OS, whether or not the external drive can be made bootable, and how you make the backup. The easy way to test it is to unplug the power connector to your internal hard drive and try to boot the machine from either the external drive and/or the CD/DVD. If you don't have a CD/DVD you'll need to make sure the BIOS will recognize the drive as bootable and boot from it.
I use a freeware program named EASUS to do a disk copy from my bootable hard drive to another hard drive mounted in an EZ-Swap2 device that lets me turn the backup drive off. That way the backup drive is only running when I do the backup and doesn't sustain any wear and tear from usage while I am not backing up...