If your sdc1 is a self-contained Linux then it can be booted up without the swap partition but adding a swap does no harm. You will need to do the following tasks
(1) Best way is to use a Linux Live CD, any one would do.
(2) Boot up the Linux Live CD and mount sdc1 which will become a different device if the original sdb is withdrawn. Edit it /etc/fstab to reflect the new partition referece of the swap and the root partition of the Linux. Forget about the UUID as you can survive without it.
(3) Edit /boot/grub/menu.lst to reflect the Linux is in the 1st partition of the ?th disk.
(4) Restore Grub
Then your Linux should boot fine afterward.