That's a standard desktop-type 3½" HDD with a SATA 3.0 Gb/s interface (aka SATA II). You certainly should be able to plug it into an external drive case designed for a common SATA II drive and it should work. The green printed-circuit board containing the broken connector is just plugged into the back connectors of the HDD unit and you'll have to remove that board.
In choosing a new external case, just make sure it has the external connection system you need to connect to your computer. (This is NOT the same as the internal connections to the HDD unit itself, which will have to be SATA II or more properly, SATA 3.0 Gb/s.) How is it doing that now? Is it USB2, or eSATA, or whatever? Look for a case that has the same connection system. Many actually come with 2 or 3 possible connections, and you use only one at a time.