Do Indian citizens living in the USA and having a Green Card for more than 10 years need a visa to visit Australia and New Zealand as a tourist for about 10 days in each country?
Australia offers ETA for citizens of several countries, including the US, but not including India. Having permanent residence in one of those countries does not suffice; you must have a passport issued by a qualifying country.
Australia also offers eVisitor for citizens of several countries, not including India, so that also won't help. You therefore need a visa for Australia.
New Zealand offers a visa waiver for citizens of several countries, including the United States, but not including India. Eligibility depends on being a citizen of one of the visa waiver countries. US non-citizen nationals are included, but not permanent residents. You therefore need a visa for New Zealand.
You can qualify for the programs mentioned above if you naturalize in the US and travel with a US passport.