What causes the water pipes to produce a high pitched squealing sound a few seconds after the toilet has been flushed and the tank has refilled?

The high-pitched sounds you hear are the upper harmonics of water hammer, and flushing the toilet without adding enough toilet paper to absorb the squealing sound. Without toilet paper in the pipes, they will vibrate and possibly crack the solder in the pipe joints. If you really want to do yourself a favor, wrap ALL the water pipe joints in high density foam rubber (HDFR - Plumbing supply shops provide these in assorted colors to mark off different areas - kitchen, bathroom, garage, etc.).