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.

find . -print0 | grep --null 'FooBar' | xargs -0 ...

I don't know about whether grep supports --null, nor whether xargs supports -0, on Leopard, but on GNU it's all good.

The easiest way to do what the original poster wants is to change the delimiter from any whitespace to just the end-of-line character like this:

find whatever ... | xargs -d "\n" cp -t /var/tmp