Yes, if it is a game for which the MySQL connection always need to be present hosting it yourself would not be practical (unless you have an old PC and a dedicated internet connection).
If the game is still in development you could try some of the free MySQL-providers and see how their quality are.
(search for
free mysql hosting with external access)
If in the end there is a lot of money involved and the game needs to "grow" it's best to choose for a professional (paid) option. Just make sure the database is accessible from everywhere and not that you need to add a host to the settings each time for it to be allowed.
B.T.W. if it's only for login and/or small communication packages (i.e. scoring etc.) you could also create a small interface in PHP on any given website. That PHP would have local access to the MySQL-database because it is hosted on the same machine. (every website provider with a MySQL-database option will do in that case)
Host gator says you can access it remotely. Does this mean this SQL database will work? It says about just adding IPs. I want it accessed to everyone. Or should I upgrasd my 000webhosting account?
Adding IP manually wouldn't be practical in your case. If that's the only access they allow it won't work. I'm not sure if 000webhosting does the same. (That's why I said above you needed to make sure they allow "wildcard"-hostname access, I'm not even sure there are any with that kind of access)