You don't want to use the pump you linked earlier - those really cheap diaphragm pumps are
1. Extremely noisy
2. Will burn out quickly. They have a pressure regulator on them so what happens is the pump brings the system up to pressure, then the pressure switch kicks the pump off, the misting nozzles bleed off the pressure, and the pump kicks back on. This repeats about once a second (or faster depending on the number of nozzles and the flow rate of the pump). Those cheap pumps just aren't built for that kind of abuse and the switch will very quickly burn out.
3. The connections on them tend to be a funky size and are a pain. They're usually made in China so while the description says it fits 3/8" hose it is more likely to be some weird metric size that is hard to track down here in the states. You can't just use any hose that you can stretch over the barb either because you're working with high pressure.
I know all this because I tried the very exact same thing as you're thinking. After dumping around $60 into my project I discovered I had two choices - dump more money into my own project for the right parts I needed or cut my loses and just buy a Mistking setup. I chose the Mistking and haven't regretted it.
You CAN DIY your own Mistking/Climist clone - you just aren't going to save enough money to make it worth it (if any at all). The first thing is to start with a correct pump, look at RO (reverse osmosis) booster pumps. Like these...
http://www.ebay.com/itm/AQUATEC-880...129652?hash=item2a732fa8b4:g:3z8AAOSw8gVYB-8v
http://www.ebay.com/itm/AQUATEC-884...885629?hash=item2a4b1e077d:g:kjQAAOSwk5FUw802
I've read that at one point in time either Mistking or Climist or perhaps both were using rebranded Aquatec pumps like in those links. Don't quote me on that, but its what I have read. In any case those are the type of pumps you need to look at. Now looking closer at the nozzle system you linked it isn't going to work in this system. Its designed for relatively low pressure, if you hook it up to either the pump you originally linked or those ones I did something pretty spectacular is going to happen pretty quick. Either the line will blow apart because its not rated for the pressure you're working with or probably more likely to fail first is it'll just blow one of the barbed connections out the end of the hose.
Nozzles are a little tricky, cheapest solution would be these. I actually used a few of these until just a few weeks ago on my Mistking setup (I was too cheap to pony up for the additional MK nozzles and I had these from my failed project).
http://www.ebay.com/itm/1-4-Slip-Lo...hash=item3ac0bdce4a:m:muvRtiTPrYJgjK9z-YSGw3g
They work alright, only thing I didn't like about them is the spray "cone" is rather tight. Whereas Mistking nozzles have a very wide spray pattern (probably 120º at least), the brass nozzles I had at least had a cone more like 60º or so. Mine kept plugging too even though there shouldn't have been anything in my system to plug them as I'm using RO/DI water.
If you want to DIY actual Mistking nozzles you need to look at something like this. Mistking nozzles ARE Tefen brand nozzles, mine have a number stamped on them but I haven't had luck tracking down nozzles with the same number (I figure its a model number but perhaps I'm wrong). I'm also not sure what the back end of the nozzles looks like as I've never had the need to unscrew one to look.If you were to use these nozzles you'd of course have to find the right connections to go from 1/8" NPT to 1/4" JG (John Guest).
https://smile.amazon.com/Plastic-No...d=1488116202&sr=8-2&keywords=aeroponic+nozzle
https://smile.amazon.com/Plastic-No...d=1488117345&sr=8-9&keywords=aeroponic+nozzle
As you can see, the parts and pieces start adding up quick. In my experience I number crunched things 15 ways to Sunday and what I came up with is you can DIY or you can just buy a Mistking/Climist system prepackaged - you end up spending pretty close to the exact same amount of money either way by the time everything is said and done.