Is there any program on Mac which can zip big files in small pieces (free ones are preferred)?
Any command-line archiving tool can do this when combined with the split command:
The files generated will look like:
To put them back together, use
If you already made an 'Archive.zip' of some files using the right click menu, you can split it after the fact by typing in a terminal:
7-Zip can do this, and it's open source and free.
You can find the Mac OS X downloads towards the bottom of the page.