Your Nest Wifi router is a router and firewall that creates a new "inner" network. Any switch should be connected inside this network (i.e., connected to the LAN port on the router). It's possible your modem also has router/firewall/NAT functionality. If it's bridged, putting the switch between your modem and the Nest Wifi Router is a bad idea, as devices connected to the switch will fight for the only available public IP address from your internet service provider, which I suspect is what is happening in your case. I would imagine if you disconnect the Eero router, and connect your Nest Wifi router it will work but explains why you can't have both routers connected to the switch at the same time.
The short answer is the fundamental technology at play here doesn't support what you're trying to do. It's also strongly not recommended to try to use mesh Wifi products from two different manufacturers. You're going to create a whole host of problems by trying to do that. My advice would be to pick one of them and have Wifi points solely from the same manufacturer.
Your modem needs to be connected to the WAN port of whatever you choose to go with, LAN port to the switch, and then the switch can feed everything else in your home, including any secondary Wifi points or wired devices you may have.
