Squarespace and weblium are not offering you to use third party web hosting service. If you want the third party hosting then try CMS (Content Management System).
You will get a website builder with your server. You can host on Joomla, or drupal WordPress. These are the most reliable website builders that can be hosted on any hosting of your preference.
Squarespace and weblium are not open source. Have you tried using wordpress, joomla, or drupal. These are the most popular website builders that can be hosted on any server of your choice. Most host like ourselves offer one click installs for these website builders through control panel.