Trackpad not registering clicks (nor a USB mouse click) on Macbook Air

Try an SMC reset and PRAM reset (hold the PRAM reset for at least three chimes).

Try Safe Mode.

Try Recovery Mode (Command + R) or Internet Recovery Mode (Command + Option + R) to see if you can click on the menus or utilities.

Option Boot the laptop by holding down the Shift key just after the startup chime so the Apple boot picker menu appears. Can you use the trackpad or mouse to click on the WiFi selector list or click on the boot drive icon to boot the laptop?

There is a chance the battery is swollen and pressing on the Trackpad and preventing it from clicking, or the Trackpad is defective. It is very unusual to have the external mouse not work and would tend to tell me it is a software issue, but the Option Boot test would prove if this is a hardware issue.

Good idea. My first thought was "swollen battery" then I saw the USB mouse click fail and thought "software" but it is possible that a confluence of bad luck brings both swollen battery and bad mouse together.