Yes, with a park hopper, you should be able to get into another park if the MK is closed. However, you may not have any choice of which park that would be.The only time I personally have been to WDW during Christmas week was in 1976 (obviously things have changed since then) so what I'm telling you is based on research rather than experience. EVERYTHING I have ever read says that the parks are jam-packed during this week. You can read more about park capacity issues
HERE and
HERE. Most of what I've read about capacity talks about the parks being closed during daytime hours, usually between 10:00 a.m. and 6:00 p.m., and does not discuss Extra Magic Hours. Usually during the times of the year that the parks might reach capacity, the regular park hours are so extended that EMH isn't really a factor.
With a park hopper ticket, if one park is closed for capacity reasons, you should be able to hop to another park that is not having capacity issues. Transportation to that second park, whether using your own car or Disney transportation, will be challenging because the roads and buses will be packed. I also researched whether it had happened that all four WDW parks had ever been closed because of capacity issues. I did find that WDW was completely closed on only a tiny handful of days in 30+ years of operation. These all-park closures were related to hurricanes or 9/11. However, I did find that on December 27, 2006, three of the four parks were closed because they were full to capacity. At one point during that day, the only park that was allowing guests to enter was Epcot. You can read more about this if you go to the Orlando Sentinel website and search for an article by Scott Powers on that date called "Packed Disney World."