This is some sort of RF test connector with a built-in switch. Possibly an MS-156 or MS-147. There is a trace going under the connector from the bottom, and another one from the top. One side goes to the radio in the phone, the other goes to the antenna. Generally the test connector is installed so that when something is plugged in, the antenna gets disconnected and the external connector is connected to the radio in its place. This allows easy testing of the radio during assembly to ensure that everything works correctly. This is necessary because if the antenna is not bypassed, the whole phone would need to be placed in a faraday cage to isolate it. My phone (a galaxy S3) has three of them under small round stickers.
They're not really designed for use with external antennas, but that doesn't mean it's not possible, just not convenient. Generally they're designed to work with pogo pins in an automated tester, it may be very difficult to find the correct adapter. And then you would need to know what frequencies the radio needs to use, and find an antenna that wil work across those frequencies. Modern smart phones have radios for GPS, Bluetooth, and Wifi in addition to the cell network radios LTE, GSM, Edge, 3G, etc. These radios operate in different bands and in some cases share the same antenna. There may also be complex RF switching to select the correct antenna and/or switch antennas on the fly, complicating the process of properly adding an external antenna. You may need more than one antenna to get everything to work correctly.
From looking at the images of the S4 in ifixit, there is definitely more than one antenna. Samsung does an excellent job integrating everything, though, so they are not very obvious. Looks like there are at least 4 antennas in the back cover - two on the top right, one on the top left, and one at the bottom. You can see two of them after taking the battery cover off; they are under the two small plastic covers on the sides at the top. Then there is another one that curves around the inside of the top right corner. Then there is at least one more down at the bottom with the speaker.