Relationship between Ethernet Switch and SQ

It all depends how deep down the rabbit hole you want to go. If you're considering inserting an audiophile switch, where currently a switch isn't actually needed, I'd say you'd have to replace the CAT 5 cables with audiophile CAT 5 cables for the synergy it may provide. You wouldn't want any improvement from the switch limited by inferior cables...

No one will be able to give you an absolute answer. I would think that any answer will probably be laced with unproven pseudoscience (like my answer).

Bottom line is whatever helps you sleep at night...

You may either invest in upgraded switches and networking or higher quality streamers. If you love fiddling with the technology you can do the internet thing, otherwise the better streamer the better the sound regardless of the network. I have and know of a number of $150K systems plugged into wifi extenders costing $59 and cheap routers.
 

Ultimately you can improve up to the intrinsic level of your streamer by fiddling with the network. However at some point a better streamer is required to go further. 

@lalitk It is an anclient Linksys EA 4200, but I have an ifi iPower supply on it and another one on the modem.  LMK is you think there are measurable gains to make in that department.  I had considered a new more powerful router for sake of non-audio network performance in the house, but I vaguely recall reading that doing so may be a net negative on audio noise.

On the CAT 8, a clear limitation I have is that I am using ancient CAT5 that runs under the floor and in the wall - probably not changeable without major impact.  But I can choose to just run nice new cable across the floor for listening - at least when my wife is not around (I do not have a dedicated audio room).  In fact, that is reason enough to add the switch! (Premium cable on the floor for serious listening, hidden cable if I want the room to look tidy for guests or whatever).

@ghdprentice  Would appreciate more guidance on the streamer.  I have always used Bryston BDPs (currenly a 2) because that is what I am familiar with from way back when.  Blunt, low feature devices, but with Bryston's reputation on build quality and the singular purpose of making and passing on a quiet digital audio signal, I have always suspected that they probably hold up pretty well from a SQ standpoint.  I'm talking AES/EBU - USB is another rabbit hole I do not want to mess with again anytime soon, and I suspect newer tech would be better.  But I have never tested another streamer.

You will likely get almost as many different opinions as responses.

Mine is, you could do a bunch of stuff and buy a bunch of stuff but, unless you have a noticeable noise issue, you will be hard-pressed to hear a difference from the changes being discussed here, particularly since you are already using the ENO in front of your streamer.    No harm trying your LHY switch just after the router and listening for whether you can hear a difference.

After reading the responses, I am curious about the difference between CAT 5 cable and “audiophile” CAT 5 cable. If I were to guess, the biggest difference would be the price.

@lalitk I should have had a little more confidence in my ability to listen critically.  The LHY, I am reasonably confident, is a net negative.  The music sounds just a tiny bit more thin and shrill.  So, nice new cable run across the floor straight into the Eno seems to be next, and curious what additional feedback I get on streamer and router.  As @ghdprentice says, I would suspect that streamer is the most important part, though I am not clear on how to best figure out how to get lift over the Bryston.  I don't care about a lot of features, just SQ.