> Is Moodle necessarily more secure on a university server?
I haven't met two people who understand the same thing under http://en.wikipedia.org/wiki/Computer_security. Same thing with http://en.wikipedia.org/wiki/Computer_insecurity. Whatever you mean by security, I don't think being "on a university" is in anyway decisive.
Limiting ourselves to Moodle servers, there are at least four levels involved: 1. hardware infrastructure 2. system software 3. Moodle software 4. humans. May be you can expand that list and give marks to the two alternatives.
> If not, how can I demonstrate that using our hosting provider (Siteground) is better than hosting on campus?
You need a http://en.wikipedia.org/wiki/Security_audit .
> And how can I demonstrate that Moodle is more secure than another lesser know LMS? Say, one running with ASP.NET or another windows environment.
What does your school plan to do with the LMS. To keep it safe or to use it?
> Any advice to get me on track to keeping and growing our beloved Moodle would be most appreciated!
So there is a Moodle and it is going to be shut down because of "security concerns"? To be replaced by what? May be having no LMS is the most secure alternative!