You can't, at least not directly. (I'm assuming you're talking about either x10Hosting's Free Hosting plans or garden-variety LAMP-stack shared hosting.) Unless you're using SQL Server-specific functionality, though, you should be able to export the data as a series of simple SQL queries that can be directly used by or easily adapted to the MySQL database server environment. (MySQL and SQLite are your only choices on Free Hosting, and SQLite isn't appropriate for even moderately large data sets.) If worse comes to worst, and if your data aren't too large or complicated, you can also export to something like a CSV file (or set of CSV files), then import into MySQL tables.
You're also restricted to PHP as a programming/scripting language on Free Hosting, so if you've developed using .NET, you don't have access to the runtime you'd need. (There are LINQ compatibility libraries for PHP, though, so translation can be easier if you've used LINQ.)