That SSR can only switch AC loads. It can never switch off DC.
Get a regular relay. Preferably on a driver board or shield.
An H bridge will work but is unnecessary. There is also a big risk that you will destroy the servo with reverse polarity.
A single P-type mosfet (one quarter of an H bridge) can work but it requires more construction skills.
I like to use high-side switching chips designed for automotive uses. BTS5012 or BTT6010 are great but a little complex due to the current-feedback features. BTS716G is good as it could switch 4 things.
Make sure that when the servo is off that you don't attempt to output anything on the Arduino pin connected to the servo. Make the output LOW or make that pin an input temporarily.