My ASUS G1S has a BIOS option for AHCI/SATA, but you probably already have it set for SATA (if your BIOS even has that option, which is useful for installing WindowsXP).
If the external drive enclosure has an optional power supply then make sure it is powered up and the eSATA is plugged into the laptop before the laptop is turned on. Hotplugging is supposed to work, but try attaching it and spinning up the drive before powering on the laptop.
Have you checked if the BIOS can see the eSATA drive? Hit ESC (or whatever your BIOS uses to get the "BBS POPUP" (that is what my ASUS BIOS calls it) menu for boot device selection) and see if it lists the external drive. Mine shows "IDE:Seagate FreeAgen" as one of the choices, even though that drive is not bootable.
If you want to test the hardware, you could download a small Linux Live image and boot it from CD or USB drive. For instance, RIPLinuX works with my ASUS G1S connected with eSATA to a Seagate FreeAgent Pro external drive. The image is about 128MB for the X-Windows version. Pendrivelinux.com has an easy way to install it to a USB drive under Windows XP/Vista/7. Just replace the RIPLinuX-10.x.iso with RIPLinuX-11.3.iso (current version as I write this) in the instructions.
Install RIPLinuX to usb from windows
Or, write the RIPLinux-11.3.iso
to a CD or CD/RW as an image.
Hit ESCAPE (or whatever key your BIOS uses to select the boot device) and pick the right device.
RIPLinuX 11.3 has an option on the menu "Boot Hardware Detection Tool!" which is a quick test. It showed two disks on my G1S with the FreeAgent Pro attached. My G1S has an old ICH8 and Jmicron JMB360 controllers, but the simple hardware detection tool might not be able to detect your newer controller.
The RIPLinuX 11.3 kernel is a very recent 126.96.36.199, so it would have a better chance of detecting your SATA controler. Choose the 4th RIPLinuX menu item with the "(skip keymap prompt)" to get 32-bit X with the default US language and keyboard. Right click on the desktop to get the RIPLinuX application menu. Select "Mount/Unmount" (third on the menu) and if the eSATA is working, then the eSATA drive should be on the list of mountable partitions (probably as /dev/sdb1 if it is the second hard drive). The last menu item is "Logout" with "Halt/Poweroff" as an option when you are finished.
Here is the homepage for (R)ecovery (I)s (P)ossible Linux rescue system