In case your chat button (or any other contact widget) is not loading, it is a good practice to check your browser console to see what is happening. In case you can see a response saying "Request is not correctly encoded" the problem is at your site. Most probably you are using a prototype.js library which is old and it conflicts with LiveAgent javascript functions.
The exact problem is that the old library is adding extra quotation marks to request parameters which causes JSON format not working correctly.
To fix the issue, you have to update the prototype.js library to a new version.