I don't believe either of the 2 bugs you mentioned will eat Morning Glories.
Can you tell if this is a clean cut at the base of your plant, or is there a dark shriveled area?

Emma

I agree with Emma that sow bugs / pill bugs will not eat living plants but like millipedes feed on decaying matter and act somewhat like earthworms in helping to convert decaying matter into humus....and centipedes are carnivorous...

It appears to be a clean cut about 1/4 inch above the soil.
A few of the morning glories are just cut and the top is still there, but most are cut and the leaves are gone. You can just see the cut stem part.
I was seeing a few slugs on the ground, but haven't seen any in few weeks since I put out sluggo.
The JMG are all in containers and the soil came from Lowes.

Ruth,
Your answer indicates it is not 'Damping Off Disease', but as you know Texas has a lot of critters that love MGs. I do know that the slugs eat on my MGs so hopefully your Sluggo will take care of that problem.

My next guess would be 'cutworms'.
http://en.wikipedia.org/wiki/Cutworm

Here is some interesting information:
http://www.dirtdoctor.com/newforum/root/cut-worms-grubs-in-my-raised-bed-vegetable-garden-t5928.html

especially this suggestion:
"To keep cut worms from cutting off individual plants simply push a small stick a couple of inches down on each side of the plant stem. a cutworm eats around the stem and this will prevent them from doing any harm to the plant. I have never lot a plant of any kind to a cutworm since using this method"

Control:
http://www.bayeradvanced.com/insects-pests/products/complete-brand-insect-dust-for-gardens

http://www.ehow.com/how_9369_control-cutworms.html

Emma