Can ATS01N2...QN logic inputs LI1, LI2 be switched on using external 24V DC supply?

Yes, it is possible. Both LI1 and LI2 can be supplied by external 24V DC. Connected following:
0V from external supply to COM terminal on ATS01
24V from external supply via external contacts (push buttons) to LI2 (2-wire control) or to LI1 and LI2 (3-wire control)

Released for: Schneider Electric USA