If you're talking about bodging a couple of RCA phono plugs on to the ends of some Cat6 then the answer is no. The digital coax S/PDIF signal is designed to run in a shielded 75 Ohm coax cable. Cat cables are neither shielded nor 75 Ohm, so the results won't be what you're hoping for.
Cat Baluns are the way. However, you'll need something substantially better than the stereo analogue (red & white) to Cat baluns. The problem is bandwidth - or how much information the pipeline will take.
If we stay with the pipes analogy for a second, stereo audio could be thought of as needing the flow you might get through a drinking straw. By comparison, digital audio needs something about the size of the plumbing pipes under your kitchen sink. Even with compression, there can be a lot of information travelling via coaxial digital audio connection compared to basic analogue stereo.
Here we come to our second hurdle.... Availability and price.
There aren't many companies making consumer-level baluns to convert digital coaxial audio (S/PDIF) to travel over Cat cable. That's partly due to a lack of demand, and partly a technical limit because a poorly designed balun can introduce jitter which can wreck the signal. The
Muxlab baluns (£35 ea, £70 for a pair) should work fine and are about the cheapest branded product I've seen. Decent bandwidth spec too at 25MHz. There's also the marginally cheaper
unbranded baluns at £61/pair. These will also carry video as well as digital coaxial audio, but whether you use that feature is entirely up to you.
Other than some Niles baluns (good gear) that are used and for sale from the States, I haven't seen anything in the right category from Ebay. Some chancer is advertising the Muxlab baluns at £67
each 
on Amazon. There's nothing else suitable oming up with a quick search, but you may turn up some options if you have time to browser about.
Now comes the final hurdle... Basic baluns don't work with your network switch.
The signal might be carried by Cat cable, but what's actually going down those strands of wire isn't a data signal. For that reason, you can't mix balun signals with data signals. It doesn't work.
The sort of baluns under discussion here turn an unbalanced signal in to a balance signal so that any noise picked up in the unshielded Cat cable during its journey can be neutralised in the unbalancing process. What this conversion process
doesn't do is convert consumer audio and video signals in to the sort of TCP/IP data packets that your network is expecting to see.
If that wasn't disappointing enough, baluns are only really happy (read as:
Reliable) when there are no joins in the Cat cable going from A to B. That means that even if you unhook the cat cable at the hub and connect the lounge socket to the cable tail coming in to your comms cupboard, then the fact that there's a joiner in place might be enough to upset the passively-powered balun signal.
There is a technology called HDBaseT which will work with some network switches; not all though. And it's recommended to have a separate AV balun network completely segregated from the data network. Oh, the other thing, HDBaseT ain't cheap.
Sorry this isn't better news, but trying to make stereo audio baluns carry digital audio - especially bitstream digital audio for 5.1 - isn't ever going to end well so you might as well hear it upfront.
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