Can I use my old laptop hard drive in my ps2?

May be...

Or do I need to get one specifically designed for the ps2?

Yes. However, that would be a difficult proposition. The PS2 disks are specially formatted out of the box, but through the use of software like HD Advance you can convert a regular DVD to work in your PS2. The older PS2's use an expansion slot, and a proprietary expansion card with a built in ethernet and 3.5" drive must be used. The real reason for why you most likely will not be able to do this is because sony does not sell these anymore. But if you have one on hand, or buy one on ebay, any 3.5" drive WILL work with modification.