What is the fastest way from Zurich to Eindhoven, by public transport?

It's a matter of personal preference, of course, but I'd probably take the train. First airplane and then train sounds like a lot of hassle (check-in, check-out, transport, etc.). Also, you could get into trouble if the airplane is delayed and you miss the train you have booked.

On September 12th, for example, there are various eight hour train connections, but then one has to change at least three times, which can be risky.

However, on bahn.de (screen shot in German), I found one connection where you only have to change twice. Granted, it takes ten hours, but trains should be comfortable (ICE) and you have one and a half hours to kill in Frankfurt, not the worst place.

Don't forget to check other days. September 12th was just a random example of a day in autumn.

One train option that can be quite economical, and save you some time is to take the night train. There is a daily night train from Zürich to Amsterdam. Take that train as far as Utrecht, and take an IC to Eindhoven from there. The "CityNightline" night train has very good deals if you buy sufficiently in advance. You leave Zürich in the evening, and you're in Eindhoven the next day in the morning.

According to Rome2Rio going through Amsterdam or Dusseldorf are your two quickest options.

NB: pedantic sidenote: There is also this option of going from Zurich (the Netherlands) to Eindhoven which would take you 3.5 hours by public transport.