Some of these files may have corrupted and are causing the downloads to freeze. Navigate to the workshop content folder (in my case. D:\Steam\steamapps\workshop\content) and open the DayZ folder (221100) and delete the contents of that folder.
Reopen Steam from the root folder, not your desktop shortcut and be prepared to login to Steam again. Press OK and be prepared for Steam to exit. Type "Run" into Windows' search and enter the text "steam://flushconfig". This should be the same drive your game is installed on.
Ensure your destination install drive has enough space. Sometimes corrupt files within DayZ itself can cause the mods to stop installing. Verify game files for DayZ using Steam client.
This is a very easy fix that seems to work for a lot of people. Even though you may have a region that works best for you, I did have mild to moderate success just simply changing region in Steam. Change download region in Steam(not my video and I apologize for exposing you to something like that. If you see your downloads stop, close everything and try to restart the download. Some files will not install unless they are closed. When downloading mods via Steam, close anything related to DayZ including the launcher, the game and DZSA Launcher. This gave me a complete list of the required mods to play on any MTAS server and comes with the added benefit of subscribing you to all of the mods on Steam. It is safe to use and will tell you what mods you need to play what servers. Anything in bold is the general gist of the bullet point and I will include links where applicable. I'm going to list a few of the ones I have seen out there within this thread to help those who may have the same issues that I had trying to get into the MTAS servers. There are many, many different fixes out there and a lot of them simply will not work.