Any difference between the one you listed versus these?
Only asking because I can't see the back of the amp switcher you listed on amazon .
This isn't really a repair issue, but we can let that slide and try and help you out.
The amp you linked is a two channel amplifier. Specifically, it has one pair of identical amplifier stages which take each one channel (one right, one left) and makes them louder. Two.
There are four sets of speaker outputs because each of the amplifier channels are paralleled so that you can have a 'front' and a 'rear' that can be switched on an off or played at the same time.
However, do not be tricked. This would be the same as if you were running a race, and I hopped on your back and made you carry me with you, then claimed I also won the race. You did all the work and if you had to run faster you probably would have collapsed.
You want to run six speakers from two amplified channels. That is three speakers on each. Not only am I on your back, but now you are also carrying my buddy in your arms like a husband carries his bride after the wedding.
In this scenario, even if you are 'The Rock' you would be at risk of collapsing if made to run even moderately fast (this amp is 100W, so not 'The Rock' and more like a pretty-fit but normal guy...).
If you parallel speakers three to a channel, you could kill the amp easily (it would probably trip the protection and just constantly turn off and on since it is a modern amp)
You can do this in another way, but it will involve a huge trade-off. You can get a speaker splitter/selector with 'impedance protection' in it. This will let you divide the signal over many speakers but will make sure that the amp never gets in the danger zone.
Of course, nothing is free -- the signal will be incredibly wimpy-fied and you will lose fidelity and dynamic range (everything will sound flatter, basically).
You could get an active mixer/splitter -- this would be a box which takes your signal and amplifies it again for distribution. At that point why not just buy an amp that has many amplified outputs instead of a two channel? It would be much better and not much more expensive. 5.1 surround is not anything fancy these days, it is pretty standard.
I would not put six speakers in the ceiling with that unit. You are much better off figuring out the room acoustics and getting a decent set of floor standing or bookshelf speakers and setting them in a position to fill the room. Ceiling speakers are completely stupid anyway since they just point down and unless you are right under them you are effectively vibrating the structure with little acoustic reward.
Get four speakers and imagine that the front of each one has a laser beam coming out of it. Set up the speakers so that the lasers meet at an X in the sweet spot of the room and at mid level where there are few obstructions. You will have 1000x times better results with this.
Here is a cheap switcher/splitter with impedance protection in case you want to go that route: