Relays make very good switches, and are the least-trouble solution. The output is isolated from the control, their conduction path is linear, their residual resistance is very low, they will withstand a high off voltage. Their main problems, slow switching and erosion of contacts during switching, are not really issues for your specific application of switching loudspeakers. Use a good old-fashioned mechanical relay if you can. ULN2003 makes an excellent relay driver.
ULN2003 is completely unsuitable for audio.
SSRs come in two types, Triac and FET. Most types are Triac, which is completely unsuitable for audio. The only FET output types I've used are very low power, designed for switching sensors, resetting integrator capacitors, that sort of thing. There may be bigger ones out there, but I don't know.
If you must use solid state, then FET is really your only choice. You need something that's resistive and linear when on, other power semiconductors tend to be non-linear. As FETs have an integrated body diode, you need two in series, back to back. You need to find some way to power the gates to around 10v to switch them on, and isolate that drive sufficiently from your audio amplifier, and from the other switches.
If you don't have too many switches, then as they don't need any power, and don't need to switch quickly, something as simple as a PP3 battery per switch would make an excellent solution. The battery lifetime would basically be its shelf life. An opto-isolator would provide cheap ideal isolation, and doesn't use any power on the output side. With these values, this switch will take a relatively long time to switch off, perhaps a few mS. No good for an inverter, but no problem for a switch.
simulate this circuit – Schematic created using CircuitLab
There are other ways that the voltage control could be delivered, a small current source from a high voltage, transformer isolated, or even ghosting off the audio power