My microprocessor has an ethernet port on it, so I guess it would be the same connection as the USB to Ethernet IC?
Don't guess, verify.
Many processors don't have a direct BASE-T ethernet connection. Instead they have some variant of MII and expect you to use an external PHY. You can get switch chips that connect directly to a MAC over MII avoiding the need for a PHY, for example the LAN9303 from Microchip.
Unfortunately I haven't seen a switch chip with two MII ports on it, so you will probablly end up needing to connect at least one of the devices to the switch over BASE-T
As to connecting two BASE-T phys without magnetics it is possible, various vendors have appnotes on it but if you are trying to connect parts from two different vendors in this way you are likely to be on your own analysing the circuits and trying to design a transformerless equivilent.
In general the transformer provides three functions.
- It provides isolation
- It converts a pair of single ended drives to a differential signal on transmit.
- It establishes the desired common mode voltage on receive.
The first is not needed for two phys on the same board but the second and third points will need to be addressed in some other way.