Use LED instead of Zener
Yes, for a USB Low-Speed applications 3-V LEDs can be used in this schematics. 

Actually, in this case these Zeners serve some USB functionality, and their main purpose is not ESD protection but voltage limiting. Reason is that this MCU  is powered from +5V supply directly, without any 3.3V regulator. At 5V the ATtiny 
I/O drivers have about 25 Ohms, and would drive hard to 5V, which is not what USB needs nor likes. The signals should be limited to 3.3 - 3.6 maximum in order to not to fry the USB host. That's why the schematics uses the insane 68 Ohms in-series resistors, and 3.6V Zeners. The data signal clipping don't need to be precise, so 3-V LEDs will be okay, and even add some fun illuminating USB traffic.

In the past the USB interface used to be tolerant to direct contact with +5V. With modernization in silicon technology and integration of USB physical interfaces (PHYs) into low-voltage 22 and 14 nm silicon (where transistors are barely 1-V tolerant), the requirement to hold 5V on data lines was eliminated. So the USB lines better be protected from DIY devices like this ATTiny projects.