I'm a little late to the party but I can confirm I too am having these clicking sounds with my U-Verse Gateway. I do not have the TV or Telephone service, strictly Internet service. A month ago the modem would click loud enough for me to hear in another room. For me, it's a one time double click per reboot. The modem then resets, drops the connection and reboots. Wait about 5-10 minutes & the entire click/reboot process starts over again.
I attempted a hard reboot (unplugged 30 seconds) and a reset (red button). Neither had any impact. My internet drops for a few hours then returns. The SSID disappears & the LAN drops as well. Yesterday, it dropped and is still down 26 hours later. U-Verse tech ran remote tests & said they would have to send a technician. What's odd is how until now would always reconnect after a few hours.
I too was surprised that a solid state device could make a mechanical sounding clicking. While it may or may not be a mechanical sound, the sound definitely exists. And having done a few searches, this is not strictly something experienced by the OP or myself.