I need to encrypt the SQL Server connection string (and a user password on my login page).
MSDN suggests using the DPAPI User Store or Machine Store to perform this.
Are either of these available on my shared account (Dev Ultra w/SQL Server Plan 1)?

If not, what other technique do you recommend?

Well, I've never heard of encrypting the connection string, but there are a few alternatives to storing it in every page of your application.

If you're using ASP, you can store such variables in the global.asa file. Other than that you don't have many options.

If you're using ASP.NET, you can store these variables in your global.asax file or even your web.config file.

If you're using ColdFusion, you can store these variables in your Application.cfm file.

Any of these solutions are beneficial to what you're looking for because even upon raising an exception/throwing an error, the code within these files will NEVER be displayed, and were designed with such things in mind. You cannot directly access or download any of these files, so you don't have to worry about information being obtained in that manner either.