In my experience with laptop drives, the bare drive itself should be all the same but for some laptops, their fittings to hold the drive could be different. You can try to take apart the fittings from the original drive in your Asus notebook and try it with your friend's UJDA750 drive. Let us know how it goes.
Regarding external cases, performance will depends on the case's chipset and your laptop specs. I see that your Asus is using Pentium M processor. I believe a minimum of Core Duo or higher and a good graphic card is needed to get full use of blu-ray playback, unless you need the blu-ray slim drive for data storage only.
Did you already purchase the slim drive?