A Linux computer is not a good switch - too slow, consumes too much energy and generates too much heat, too consumes too much space, few ports. Simple 4 port switches cost 20$ new, and you can probably get used ones for practically nothing. The cost of the switch will be offset by the saved electricity soon.
The Linux bridge use case is connecting virtual machines to the outside world.
Having said that, you will have to create the bridge and add both eth0 and eth1 to it. This means that the physical interfaces will not have their own IP address anymore; the bridge will have the IP address.
Here is a description how to do that in Debian:
https://wiki.debian.org/BridgeNetworkConnections
And in Ubuntu for connecting VMs:
https://wiki.ubuntu.com/KvmWithBridge