Can I store RegExp and Function in JSON?

You have to store the RegExp as a string in the JSON object. You can then construct a RegExp object from the string:

// JSON Object (can be an imported file, of course)
// Store RegExp pattern as a string
// Double backslashes are required to put literal \ characters in the string
var jsonObject = { "regex": "^http:\\/\\/" };

function fun(url) {
    var regexp = new RegExp(jsonObject.regex, 'i');

    var match;

    // You can do either:
    match = url.match(regexp);
    // Or (useful for capturing groups when doing global search):
    match = regexp.exec(url);

    // Logic to process match results
    // ...
    return 'ooga booga boo';
}

As for functions: they should not be represented in JSON or XML anyway. A function may be defined as an object in JS, but its primary purpose is still to encapsulate a sequence of commands, not serve as a wrapper for basic data.