What's the name for this type of flowchart?

It's called a railroad diagram. It can more specifically be called a syntax diagram. Not sure about Visio support but other tools can get the job done: What is a good tool for creating railroad diagrams?

_{(source: antlr.org)}

If you have a grammar file you can try to get it working in ANTLR Works, which is specifically built for working with the syntax and grammar of languages.

These are called Syntax diagrams (or railroad diagrams).

They can be created with Visio or Creative Docs .NET

See here for more info:

Tool for generating railroad diagram used on json.org