Can I power my shift register with a 12V external power supply and connect only the data, strobe and clear pins to the Arduino digital pins?
No. CD4k devices reference their high and low voltages to a (mostly) fixed ratio of Vdd, so there will be no way for the Arduino to operate the device.

Consider using a TPIC6C595 instead, which combines the shift register and the high-voltage driver into a single device.