You're probably just seeing a lower resolution preview in whatever you're using to create the document. That's normal — applications like InDesign often show a lower resolution preview on screen, but print at full quality.
Should I make an A4 size 600 DPI image in photoshop and gimp and then add screenshots to them and then optimize them?
No. That's actually the exact opposite of what you should do.
When you take a screen shot, one pixel on your screen is represented by one pixel in the image file. This is great.
If you scale the image up, you won't gain any quality, all you'll do is increase the chance of damage, depending on the method and size you scale to. And you'll also increase the file size, making the images harder to work with and transport. The best way to maintain the quality of a screen shot is to keep it at its native resolution.
I'd advise you to do some test prints, but the best way to handle screenshots for use in printing is:
- Convert the image to CMYK if you're using full colour printing.
- Do not scale the images, under any circumstance.
- Apply any colour corrections you may need (levels, curves etc).
To print at high quality, you may know or have heard that 300DPI (or higher) is often the target resolution. There's several aspects at play, with regards to screenshots.
300DPI+ for a photo is great. This assumes you don't want to see the pixels the photo is made from.
300DPI+ for a screenshot may not be great. This is because most computer screens are in the 100 to 140DPI range. When you look at a computer screen, you often can see pixels. Don't be afraid of pixels from screenshots when you print, either.
"DPI" is just a tag associated with the file. What matters, in terms of quality when printing photos, is the resolution vs the final print size. If you make an image smaller on the page, its dots per inch (DPI) is increased. Does that matter for screenshots? No. Because you don't mind seeing pixels.
If you're after a glossy product screenshot and you don't want to see pixels, then your best bet is to recreate the screenshot at a higher resolution. It doesn't sound like that's what you're after though.