Yes you can run Mint on an external drive.
Download and burn the latest stable version of Gparted. Use it to shrink something on your first internal drive so you can make a small ext2 partition of about 64 MB. Make a partition for your swap, root, and home on your external drive.http://sourceforge.net/project/showfile ... _id=271779
Reboot with the Mint install disk and use manual partitioning when you get to that point. Assign your boot partition to the internal drive and the other partitions on the external drive. Let it install grub to the mbr of the first internal drive.