Must Israeli citizens who do not reside in Israel pay for the nationwide Israeli healthcare system?

Great answer from Jack Klaber:

No.

You can only be a member of one of the nation’s healthcare companies (Clalit, Maccabee, Meuhedet and Leumit) if you are a resident of Israel.

There is however an exception for Israeli students studying abroad or expats. During the first five years abroad they can pay monthly the minimum healthcare payment (which is today if I’m correct 175 NIS) so that if you come home to visit you are covered and can enjoy all the services of the Israeli healthcare system. If you are working abroad and send your pay slip to the National Insurance Office (Bituach Leumi) you get a NIS 40 reduction in monthly payments)

After 5 years you cannot be insured anymore and upon return to Israel need to insure yourself and your family privately for half a year until you can join one of the nation healthcare companies again. There is an option to forfeit the 6 months waiting period by paying upfront some 12,000 NIS to the Nation Insurance Office.