Was running with AntiX 14R-alpha1, and did an apt-get update && apt-get dist-upgrade yesterday about noon, then ran the laptop for the rest of the afternoon then shut down. On trying to boot up this morning I was greeted by:I'm at a loss about what to do to trouble shoot and fix the issue. I've never had much success with grub rescue mode. Can someone offer some suggestions?

That appears to be a missing grub2 file.
(On MX14, I have that file.)
Try looking for that file using a 'live' distro, preferably antiX 14, then you should be able to copy it over.

Well, seems there is more missing/corrupt than just the grub file. When I tried to mount the boot drive with a LiveCD I get an"unknown file system" error, and the drive won't mount.

Sounds like a broken hard drive (mechanical)

I got the same error when I accidentally drop kicked my 1 Terra Byte external Hard Drive.

You might wanna check it out in a live Gparted session and see if you can mount it in gparted.
Don't format or anything, Just check if it cooperates on mounting or not.

Command line mounting also can be used to check running in a live session.

I think i would also run a filesystemcheck from a liveCD. I am not that sure, but the command"fsck" might be the one you are looking for. I seem to recall gparted has an option to check filesystems too.
If the filesystem is broken but can be repaired with a filesystemcheck, and if you still can't boot, then using"chroot" might be usefull to repair the broken system, e.g. by reinstalling grub (It might also be you simply have to copy a file from the live CD, like proposed above).

I am just shooting in the dark, saying what i would do in such a situation.

It looks as though you had an upgraded from legacy grub to grub 2 and forgot to run the transfer command that apt displays in order to migrate your legacy grub configuration to the grub 2 configuration. If this is the case it is probably easiest to boot a live cd and install grub again. This can be done straight from the live cd or through a chroot as nadir stated. I think that the chroot would give you the best result as it would be specific to the install, and you seem to have a working step in the mbr.
So I would try
Boot live cd
Open root terminal
Type : mkdir /mnt/install
Type: mount /dev/sda1 /mnt/install
Type: chroot /mnt/install
Type: the magic upgrade command (sorry I cannot recall it, maybe Google legacy grub to grub 2 debian)
Type : update-grub
Type : exit
Type: umount /mnt/install
Type:exit
And reboot without the cd.
If it still doesn't work than a grub reinstall will need to be done. Much is the same as above but there are a few extra steps... seeing how it is grub 2 by the looks of things you may benefit from booting the mx14 cd if the computer is up to the min specs and using the grub repair utility from there.

Thanks for all the suggestions guys. Dave, here is what happens when I followed your idea: And, rokytnji, same response when attempting to mount the drive with gparted.

@nadir...good idea about running fsck, it reported too many problems for fix. I answered yes fix, to a few hundred file system problems, like illegal blocks and such, but finally just gave up. Must not have shut down cleanly, and scrambled things really well.

Tomorrow I may just format the HDD and do the frugal install I've been thinking about for a while.

Wow ok, I thought it was just a problem with grub being corrupted after an update. It sounds to me that rocky is right the hard drive took a dive to the mat. Perhaps if you boot live cd then you can check the drive status / health with smartctrl -a /dev/sda. If there are a lot of errors / failures / remapped sectors then that would conclude the case