Is there any way to open an ArcGIS map package (
*.mpk) file with MapInfo?
I do not have access to ArcGIS Desktop.
Inside an MPK you'll find an MXD and a fGDB. (You may find shapefiles, tiffs or other files depending on how the MPK was created, but odds are you'll find data in the fGDB).
With this knowledge, if you have an application other than ArcGIS that can read MXDs and fGDB, you can simply unzip an MPK (using something like 7zip) to get at the contents.
As for a program that offers native support to read an MPK that isn't Arc* -- I haven't heard of any.
Change the file extension from .mpk to .zip. Attempting to open the file should automatically unzip the contents.
This method works for Mac OS if the default Archive Utility did not work.