Are there standard colours of squares or lines printed on a paper? Or I can choose gray, green or blue CMYK colours and ask printing company to print squares in my particular colour?
You can use Cyan or Black or any pure cmyk (not a mix of cmyk), it should be the same price. You can choose a Pantones too if you want something like green, the cost shouldn't be a lot more.
No there isn't really "a" standard, some lines use a light blue, some are dark blue like your sample. It's up to you. But if you want to get close to the most common color for this, you can use 35-40% of Cyan only. (example below.)
If you do the design yourself, don't make the lines too thin or they won't look super sharp because of the angle of the cyan on the plate; you should even tell this detail to your printer so he changes the angle if necessary, probably to black.
Example of result you don't want if you make lines in 1-color CMYK not at 100% (middle part):
(what I mean by the angle in case I'm being confusing: http://the-print-guide.blogspot.ca/2009/05/halftone-screen-angles.html)
I guess your lines should be around 0.75pts thick but you can have them thicker too if you prefer.
If you use a Pantone you don't need to worry about the angle, and you can use your color at 100%!
The values on lined paper most likely straight CMYK for convenience. For instance the blue is most likely straight cyan 100%. But yes, you can tell a printer CMYK values or a Pantone number. Additionally if you're not sure, you can bring in an item and they will be able to match it as close as they can.
Well. I disagree a bit with the other answers so here I go.
Depending on the process the sheets are printed you can not vary the thickness of the lines for example. Sometimes this sheets are printed from a roll, which is the cheapest way of doing it, and you use some kind of needles (Imagine the ones you see on a sismograph)
Yes you could ask for a diferent kind of ink but you need to print some good amount of tons of paper for that.
This ink can be somehow diluted so you can have more consistent flow and uniformity.
But let us asume that you want to print less ammount of sheets, not full rolls.
Ok, then you choose a flat sheeted offset print.
You do not want to use a 4 head printer for that (the ones used for full color print or color selection. On a 4 head print you probably need to stick to cyan because it cost money to clean the cyan head to change the ink, and they will charge you the 4 passes even they do not use them all)
You ask it to be printed on a 1 head or two head machine. Depending on the type of printhouse this can reduce the cost to about 1/4 or half the price without considering the paper cost.
And on thoose machines you can simply use any color you want. Gold, aluminum, fluorescent ink, chocolate mousse... If they have it on stock, there is a chance most colors will be on the same price range. (Metalic cost a bit more but I do not think you want to use gold for that)
So you now can choose not only the color but also the design.
You are not bounded to use cyan. I stated that already. But you do not have too many options.
Traditionally cyan, blue and green ink could not be photocopied on an old fashion xerox machine. On a digital copier it does not matter anymore.
But imagine the color options you have. You could use any rainbow color but normally thoose looks like notebooks for kids.
Red, purple, pink, magenta, yellow orange, are too happy.
Green could be not friendly. Sometimes is related to humidity on paper.
This leaves you to some shades of cyan-blue or gray. You do not want the lines to be too dark so they are confused with the writting itself asumming black ink.
Just take that into account.