First I would go with warm white lights not the cool white, I think it makes a softer look to the bedroom.
In terms of the switching, yes a push button switch would work fine, you just have to make sure that the switch is within spec for the lights used. I have rather opted for a dimmable solution with my childrens beds. since they like to have some light at night. a good option here is this:
Its rated up to 8 amps, so if you look at the strip in your post, it consumes 2.5amps per 5m section used. (provided that you buy a transformer that can supply up to the required amps (ie 2.5amps per meter) -- you can buy a smaller amp transformer, but the lights will not be optimally bright at the highest level.
Your switches are only rated 3 amps, which means you can run about 5 meters of your spec LED Strip. -- Also to note, I would go with a stronger LED strip, generally 1 amp per meter is a nice powerful light (then you would only need a meter per child)
Another thing to consider, is the use of the 3color led's whereby you can have a color chooser.. but that may be getting too complicated.
Your series vs parallel question: You must definitely hook up the two lights in parallel, but just accomodate for the increased amps which will be required. The lights will draw the amps needed so you cannot really blow anything (its the volts which will blow the lights)
Finally: I find that the higher power led strips get a little warm, (meaning that the sticky tape sides start to get less effective over time). would be best to stick the strip to a metal surface if you could choose (it would act as a heat sync)
I actually ended up using a Chinese paper lantern to put the led strip in, it looked quite nice.
There's no code problem having a light on 24 hrs and switched at the breaker, but the breaker must be switch rated. Most new breakers are rated as such. You can check on one side of the breaker there will be a circle with the letters SWD stamped inside of it.
Best Answer
Ecnerwal make an excellent statement; doesn't matter.
I've got 25 years with the power company designing underground systems for high rise buildings, residential subdivisions and downtown areas. We hate radial feeds because if you have a problem, everyone's out. So I'd be running both sets to the switch. If something goes wrong with a section, you can disconnect that section from the switch and power up the other set so you have lights to fix the problem. (I know, I'm being ridiculous).