My program uses its own binary file type, so I assume I can't use MIME type text/plain, as it is not a 7-bit ASCII file.
Should I just call it "application/myappname"?
I'd recommend application/octet-stream
as RFC2046 says "The "octet-stream" subtype is used to indicate that a body contains arbitrary binary data" and "The recommended action for an implementation that receives an "application/octet-stream" entity is to simply offer to put the data in a file[...]".
I think that way you will get better handling from arbitrary programs, that might barf when encountering your unknown mime type.