How can I use xargs to copy files that have spaces and quotes in their names?

You can combine all of that into a single find command:

find . -iname "*foobar*" -exec cp -- "{}" ~/foo/bar \;

This will handle filenames and directories with spaces in them. You can use -name to get case-sensitive results.

Note: The -- flag passed to cp prevents it from processing files starting with - as options.