Likely, your cable modem is your router (internal/integrated DHCP server). On most modems, though,(with a single ethernet port), you will need to add a router before a switch. (of course, if you add a router, you may no longer need the switch unless the number of wired devices outnumber the LAN ports of the router).
Additionally, many ISP modems only allow one wired device to access the internet at a time (due to a single public/wan IP address allocation) when using a switch directly connected to the modem. (multiple devices unable to share/compete for this single address in this setup). The router allows multiple devices to share this single IP address. The switch expands the number of wired devices when plugged into the router.
However, just to verify. What is the model number of the cable modem, the ethernet adapter you are using, and the Roku Express (from Settings/System/About).
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First inclination is that the ethernet adapter may be part of the issue since it fails to work at both the switch and the modem location with the Express. Generally, need to use a USB 2.0 10/100 fast ethernet adapter with the required chipsets AX8877X or SMSC95XX. (there are a few users that have reportedly gotten some 10/100/1000 adapters to work, but hit and miss there.)
Update:
By using longer power-down times for the Cable Modem in preparation for trying each device and waiting for the Modem to start up fully before connecting the ethernet device, I was able to get both Roku's to find the ethernet (& Internet) - although it seemed to take two tries for each Roku to "find" the ethernet. (Not sure what that was all about). Anyway, I was able to get as far as doing the firmware updates for each Roku. So, it appears the problem is that the Cable Modem can "recognize" only one connected ethernet device at a time - even with a Switch in place - so apparently some sort of Router is required with this Cable Modem. The unfortunate consequence is that I cannot activate the directly-connected Roku devices because I cannot connect a laptop (to receive the activation code eMails) and the Roku device being activated at the same time. So, apparently, I'll need to get the Router situation settled before proceeding with activation/testing. Frustratingly, the only ISP-compatible routers have just four ethernet ports - and I'll eventually need to connect six devices at the same time. Hmmm... Seems to be a gotcha at every turn...
