component.jsx

UserProfile = React.createClass({

    componentWillMount() {
        loadFilePicker()
    },

    filepicker(e) {
        console.log("filepicker event",e)
    },

    render() {

        return (
            <div className="mdl-grid">
                <div className="paper padding mdl-cell mdl-cell--4-col">
                    <input type="filepicker" onchange={this.filepicker}/>
                </div>
            </div>)
    }
})

filepicker.js

// Copied from https://github.com/nate-strauser/meteor-filepicker-plus

loadFilePicker = function(key, callback){
    if(typeof filepicker === "undefined"){
        if(!key && Meteor.settings && Meteor.settings.public && Meteor.settings.public.filepicker && Meteor.settings.public.filepicker.key)
            key = Meteor.settings.public.filepicker.key;

        if(key){
            // Functions to run after the script tag has loaded
            var filepickerLoadCallback = function() {
                filepicker.setKey(key);

                if (Object.prototype.toString.call(callback) === "[object Function]")
                    callback();
            };

            // If the script doesn't load
            var filepickerErrorCallback = function(error) {
                if(typeof console !== "undefined") {
                    console.log(error);
                }
            };

            // Generate a script tag
            var script = document.createElement("script");
            script.type = "text/javascript";
            script.src = "//api.filepicker.io/v2/filepicker.js";
            script.onload = filepickerLoadCallback;
            script.onerror = filepickerErrorCallback;

            // Load the script tag
            document.getElementsByTagName('head')[0].appendChild(script);

        }else{
            if(typeof console !== "undefined") {
                console.log("filepicker-plus - tried to load but key not supplied");
            }
        }
    }
};