My hosting provider ask me to use xCache instead of the built in Drupal cache system Just that I have a problem with a website (Drupal 7), the server falls down every night. 
My hosting provider ask me use xCache instead of built in Drupal cache system.
Does it make any sense?
It makes perfect sense to use PHP opcode cache, you should be using one if not already using. I'd recommend APC for PHP version 5.4 and less.

xCache is somewhat similar in the way it handles caching like APC, but the mechanism is different. You have to do quick research which one suits your needs and hosting compatibility. 

If you are using PHP 5.5+ then it comes with in-house OPcache. Personally I haven't tested but when the package is bundled with software core it certainly grabs attention. You should check the use case for you. 

eAccelerator is also nice one, I used to use it few years ago, but, when I got to play with more RAM, APC is beast.

The later part of your question says about 


  use xCache instead of built in Drupal cache system


which is completely false, xCache or any other opcode cache is not replacement of Drupal Cache, it should be used together. Opcode cache is for PHP, it doesn't have anything to do with Drupal.

As for the part of replacement concerns, you can use Memcached instead of Drupal cache that is stored in database.