Q: Can a switch be used for extending the network with ethernet?

But what I was saying is that my base station starts dropping the Internet connection only when I plug in one of Extenders by ethernet.

And, when you do this, do you understand that it is up to the Fios modem/router to provide a valid IP address and other network settings to the extender(s)?  The
"main" AirPort Extreme base station should have has no active role in this.....since it is in Bridge Mode, it simply passes through the network information that is being provided by the Verizon router.....assuming that the AirPort Extreme is setup correctly in Bridge Mode and it is operating correctly.

What might help is to have the Verizon modem/router provide a fixed or Static IP address to each one of the AirPorts that you have on the network.  For example, the "main" AirPort Extreme might receive a fixed IP address of 192.168.1.10 from the Verizon router, one extender receives 192.168.1.15 as a fixed IP address and the other 192.168.1.20.  As it stands now, things are pretty much left to chance as to whether the extenders will receive a good IP address.

If you decide to set up fixed or static IP addresses, be sure to temporarily move the extenders so that they connect directly to the Verizon router to set things up. Then, once you have checked to see that each extender has received a good IP address that way, move the extenders back to their normal location. If the AirPort Extreme is working correctly, the switch is working correctly, and your cabling is good, then the extenders will receive the same settings that they did when they were connected directly to the Verizon router.

Do the extenders function as expected when plugged directly into the base station? Is the Verizon router or Base Station your DHCP server?

This sounds as if you have configured the "extenders" to connect  using wireless.....and then you are trying to establish an Ethernet connection to the extender.

You cannot do this, since it will create a huge feedback loop, which effectively crashes the network.

The solution, is to reset both extenders back to factory default settings.

Then....establish the Ethernet connection first......before you attempt to set up the extender

Now when you set up the extender using the setup "wizard", the extender will receive the correct settings from the setup utility

Post back if you need a few tips on getting the extenders set up to "extend using Ethernet".

Since you indicated that you have already tested the Ethernet cabling, you can set things up this way:

Power off one of the AirPorts that you are using to extend the network so that only one is powered on

Hold in the reset button on the back of the AirPort for 7 to 8 seconds, then release

Allow a full minute for the AirPort to restart to a slow, blinking amber light status

Connect the Ethernet cable from the switch to the WAN "O" Port on the AirPort

Click the WiFi menu at the top of the Mac's screen and look for a listing of New AirPort Base Station

Just under that listing, click on AirPort Extreme

When you click on AirPort Extreme, that will get the setup "wizard" up and running. It will take a few seconds to analyze the network and then display a screen that looks similar to the illustration below except that you will see your devices pictured along with the name of your network.

Type is short simple name that you want to call the AirPort that will extend the network

Click Next

Now watch the next screen very carefully to confirm that the AirPort is being set up to extend using Ethernet.  This confirms that the setup wizard has picked up the Ethernet connection and will apply the correct settings to the AirPort.

When you see the message of Setup Complete, click Done.

Now, set up the other extending AirPort exactly the same way, except use a different device name to keep things straight.