I'm sorry that I can not give a definitive answer to this as I have not tried it myself. I've used CMC as a thickener in "goo" for instant products, and for spreadable process solutions. I've also used it in gelatin as a thickener. In all of this, I've found, under my conditions, that the CMC redissolves when wetted if it dries down. So, all I can give you is theory here.
If you have (as you do in CMC) R - COOH (this is the symbol for an organic acid) and it meets with a Ca++ ion (or other ion with multiple plus charges), you can get R-COO Ca OOC-R + 2H+ which is a cross linked (hardened) molecule where R = the rest of the CMC molecule or molecules.
At present, I have no CMC in the lab so I can't test things and give a better answer.