Spider III makes a high pitch sound when I plug my cable into the input slot. It won't stop and it keeps echoing.

If your guitar is not connected and the cable alone causes the oscillation, I would certainly question the QUALITY of that cable.

Poorly sheilded cables can pick up ground loops.

If you short the contacts of the cable at the guitar end and that stops the oscillation, I would try a higher quality instrument cable. MAKE SURE the cable you are using is an INSTRUMENT cable and not a SPEAKER cable which I can guarantee would cause the problem.

The Line 6 amps are NOT noted for reliability so I would NOT let that oscillating conditipon persist for any length of time.

Do NOT use excessive gain at the front end,,, use higher master volume and less front end gain if you are having ground loops.