And finally, eggshells do not really compost at all. They get broken up but if you leave them alone, they will last for many many years. If you want to release the calcium they contain, which is good for preventing end-rot on your tomatoes, you need to crush them up and then add some vinegar and water. Let that bubble up for a while and then further dilute it with more water and use it to water your plants. You can throw those eggshell pieces into the garden as well, since they're partly dissolved at that point they will decompose much faster. And they will not prevent slugs or snails from eating your plants. That's another old wives tale.
Egg shells and other calcium sources do not prevent blossom end rot in most cases. Several Universities have done their own studies and experiments along with other sources that found soil tests of locations with end rot including tomato gardens were rarely deficient in calcium. The problem is calcium uptake.
One issue is the use of epsom salt in gardening. Magnesium prevents the use of calcium in plants and most soils aren't low in magnesium. Epsom salt is magnesium sulfate and even the amounts used to try to benefit plants can reduce calcium. When people try to use it in DIY herbicide mixes at high concentrations it's practically a guaranteed problem for some plants. Along with some concerns of run off. Magnesium doesn't get bound up in the soil so high concentration areas will steadily rinse out to other areas and eventually the water table or storm drains if applied frequently or in high concentrations. Overall it is bad unless used carefully and preferably accompanied by a soil test.
Excessive ammonia levels will also reduce calcium usage and can also be the fault of gardening practices in an attempt to improve plant growth. Heavy applications of high nitrogen fertilizer has multiple negatives that can contribute to end rot. A lot of rich compost is actually a bad thing in many situations. Although not as bad as attempts to use concentrated fertilizers whether commercial or DIY options when you don't actually know your soil conditions or time their usage correctly.
Plants also need a good root system and enough water to move calcium. Concentrating on fertilizing for top growth, flower, and fruit production instead of making certain the plant puts energy into a large root system can reduce nutrient uptake. Diseases like end rot can occur from improper fertilization and even high nitrogen compost applications. Any resulting food items are also lower in useful nutrition.
A final potential cause is that failure to maintain adequate water will reduce all nutrient usage including calcium needed to prevent end rot.
Unless you've tested your soil to know otherwise most ag extensions, university horticultural programs, and botanists point to other gardening practices and improper fertilization or management of plant growth and soil conditions as causes of end rot. Not a lack of calcium in the soil. High magnesium, high ammonia or nitrogen, high potassium, high salinity, rapid growth, drought, low humidity, and excessive heat are known to contribute to end rot by interfering with calcium uptake or transport throughout the plant. Since all stressful conditions, negative impacts on root development and improper water can potentially lead to end rot there is no single solution for it.
It has been proven calcium can be absorbed by the leaves and fruit of tomatos to target the problem area and potentially bypass whatever issue is reducing how much calcium reaches the fruit. Most end rot sprays attempt to deliver calcium more directly into the needed area. Overall though you need to look at all the possible causes to determine what you are doing wrong or what is wrong with your soil.
Egg shells are a more balanced source of minerals with actually quite a few different minerals in them. Including them instead of more concentrated sources like epsom salt and chemical fertilizers can avoid over fertilizing the wrong minerals and reduce end rot. Only if you don't also overload your soil with other sources of those minerals and ammonia or nitrogen containing fertilizers.
Egg shells do provide some food to beneficial soil organisms and can increase microbe population and diversity in compost bins and soil. They contain about 5% organic matter and the rest minerals. Otherwise they wouldn't be so prone to attracting pests and a potential concern for salmonella even when the egg contained inside has been put to other uses. A hot pile should eliminate salmonella risk but those with small piles or turners are sometimes advised to at least make sure all shells are heat treated before crushing and adding to compost containers.
Majority of the shell is very stable calcium carbonate so after the organic part is eliminated rapidly the rest of the shell will slowly dissolve rather than decompose. It does help to make the pieces smaller and keep compost piles from excessively drying out.
Egg shells also aid worm digestion (I never looked up why) and have been shown to sometimes deter slugs and snails so may be of more benefit applied directly around plants in garden soil than add to your compost bin.