Yes, a 2.5" IDE external case should work with the hard drive from that Toshiba. Open the system up to double check but I believe the hard drive is just a fairly standard 2.5" IDE drive. It is a bit thicker than many enclosures expect so make sure the enclosure can safely run without the cover.
I have a hard disk from a different Toshiba system (T4x00) from about the same time frame and I think some of the T1960 got identical 320MB drives. I had no problems getting the drive to work with a cheap USB adapter so I expect you to have the same result.
Well, the above is predicated on the system being a 486 T1960 not some other system that Toshiba also used a 1960 model number for.