You can use Linux or Windows hosting irrespective of the operating system on your computer.
You can see a comparison at
In short, you need Windows hosting only if you require ASP/.net/MSSQL/Access support. For most purposes, Linux hosting is sufficient and also has better support for PHP/Perl/CGI/MySQL.
A static html/flash/graphics site can be hosted on either Linux or Windows. In general, we recommend Linux hosting for most sites.