At this time of year, and over winter, Chrysanthemums outdoors are bare, with no leaves and no flowers; because they are perennial, they start shooting in spring, ready to flower later on in the year. The plants you bought will have been forced, grown under particular conditions to make them grow and flower now, but this means they are susceptible to cold and bad weather conditions during winter generally. From the images, it's clear one is in better condition than the other, as you say, but this might be accounted for by slight differences in environment or where they are positioned. It's possible one catches cold winds, or has less sunlight, or is in a colder spot than the other, and this might account for the difference.
There does not seem to be any obvious problem with fungal infection or insect infestation, but inspect the failing plant closely, under the leaves and all the stems, for any signs of problems. Your addition of eggshells will not have had this impact, or much impact at all, so that won't have caused a problem. There may be a problem at the root, but if the soil you used was new, sterile potting soil, any problem is likely to have been in the soil of the plant already rather than in your soil. If you are desperate to save it, you could dig it up,inspect the root ball for problems and if none is visible, pot it separately and keep it somewhere a little warmer, removing any dying or wilting flowers; alternatively, wait and see what happens. Both plants will cease flowering fairly soon and if the weather is cold enough, may die back to the roots, but should hopefully produce new shoots later on. Note the regrowth will be much taller than the height of the plants as they are at the moment.
