Is the other HP laptop also running Windows?
By chance, did you attempt to use any encryption software that may have been included with the drive? Or attempt to encrypt using other software?
When you enter "My Computer" on the laptop with the issue (My Computer is found now under the Start Menu> "Computer", or you may type "Computer" into the start menu search to locate it), is there another device displayed that is the drive you are looking for?
Some drives come with multiple partitions, one for storage and one with special software. If the software drive is empty and does not have formatting, or if it is an unrecognized format, Windows will prompt you to format that part of the drive when it is connected.
If you happened to format the drive with a different file system (EXT in Linux, or even TrueCrypt, for example) that would also cause issues.
One possible solution if it is a dual partition issue:
http://superuser.com/questions/226307/prevent-windows-7-prompt-format-usb-hdd (delete the drive letter, more of a "quick fix" than a solution).