Probably because Slackware uses LILO boot loader. I never use LILO but Grub. Your problem has something to do with bootloader or mapping of drive.
Try use a linux live cd (Mint or any of choice) then "chroot" into your Slackware installation. Put on your Slackware install CD and go to folder /extra there you have a tgz or txz file for Grub bootloader: you must complete the installation of grub at this point else same problem recurs. Here is a good grub 2 tutorial
However if you know about Lilo you may confront the problem from there, I assume lilo is capable of that. (I have no knowledge about this boot loader.)
Another way is to save your important files and install Slackware from the hard drive, it only takes about 5 Gigabytes of space to do that, you may resize your current windoes installation to provide that needed space.
Finally, you can install Linux Mint on that external drive: Mint manages the bootloader configurations (at install phase) before it goes to reboot. This way you can save your effort. Install Slackware later when you have become better familiar with booting process.