1) Your Yamaha AV receiver is pre-full HDMI implementation, so the two HDMIs in are just video switching. The set requires a parallel digital audio cable or analogue cable for sound. I think the model dates from around 2007/2008 when HDMI and Blu-ray were just being established. The RX-V361 was I think the entry-level model. The 461 was the step up unit.
2) You haven't said the make and model of your speaker kit. What is it?
3) Does your son's Sony stereo sound better to you playing from the squeezebox or is it playing something else such as CDs or MP3 rips?
4) What file types are used for your rips played on the Squeezebox; Format, compression etc?
5) What is the make and model of the sub if different from the surround speakers?
Until we have answers to all the above then it's difficult to give absolute recommendations, but as some general observations these are some points to consider...
The limit of the sound performance in your main system might be the AV amp, but equally it could be the speakers. Putting a good stereo amp in to feed AV speakers as opposed to Hi-Fi speakers might not give you the jump in performance you're expecting. Things also start to get more complicated if the subwoofer doesn't support simultaneous connections from the LFE and High Level inputs. (Just having both sets of sockets doesn't automatically mean that it can handle both connections at the same time. There has to be a facility to mix the two inputs within the crossover.) This is why it's important to know all the kit model numbers. A speaker switch is one solution, but it is not without its own issues too.
Next, much depends too on the quality of the digital music playing via the Squeezebox. FLAC is reputed to work best. Is that what you're using? The power supply to the Squeezebox is also known to be a weak point. Have you upgraded yours?
Even with an upgraded PSU I think the analogue outputs are still a weak link. So if you buy a stereo amp then it will have to be one with a built-in DAC to make use of the SB digital output. Either that or buy a separate DAC. Whichever way you choose then the cost is going to be relatively high (entry-level Yamaha Hi-Fi amp with digital inputs: Yam AS301, roughly £250. You'll spend similar total on a basic stereo amp and basic outboard Hi-Fi grade DAC.) You'd still have to buy a speaker switch box and do something about the sub connections too.
If it was me then I'd be looking at replacing the 461 with a better quality used AV receiver or AV amp. This
Yamaha RX-V1800 would have been perfect, and at it's final selling price of £150 then IMO a bit of a bargain too.
An AV receiver or AV amp might not be the final word in audio quality compared to a similarly priced Hi-Fi amp when new. But in the second-hand market these older receivers can and do cut it against the sub-£200 Hi-Fi amps that are probably going to be on your radar. In my experience a good £500 AV receiver coupled with very good speakers has the ability to easily show up the difference between music played from a Blu-ray player versus a dedicated CD player. You're also probably going to be able to distinguish between the system playing in stereo mode and in Source Direct mode too.
Changing the AV receiver means that most or possibly all the other issues go away too: There's no speaker switch and extra wiring to buy. There's no jiggling around with the sub connections swapping them over each time if it doesn't support dual simultaneous inputs. You will add compatibility for HD audio via fully operating HDMI sockets. You might even have some features such as cross- and up-conversion. One other
big advantage is with the higher-grade AV Receivers you'll often find they have a set of full 5.1/7.1 pre-outs. This opens the door to proper integration of an AV Receiver with a Hi-Fi amp at some point in the future.
