No they can't be saved. If they were ripe you could cut off the bad part and use the rest. Egg shells aren't going to help. Keeping the moisture level consistent is the most important thing but you might consider adding lime or gypsum to the mix.
I would cut the fruit off and toss them.

Agree. Once the BER starts on the bottom of the fruit it cannot be saved. Pitch them.

Crushed egg shells 'might' be of help a year from now if you planted in the same bucket since it takes them that long to break down but it isn't of any benefit for the current plants. Check out the FAQ here on BER and its causes. As Taz said it is consistent soil moisture that is the key, not frequency or amount. Maintaining consistent soil moisture in a 5 gallon bucket is difficult if not impossible to do.

Dave

I heard that epson salt will help you not to get bottem rot I use it and never have had bottem rot again might just be a old wifes tell but it seems to work for me.
Angie

Epsom salts contains magnesium NOT calcium. It will NOT help BER.

There is one use for BER tomatoes if you are growing open pollinated tomatoes. You can save the seeds to plant next year. Since BER is not a disease it will not be passed on to the following crop. I agree that the egg shells will be of no benefit, at least not this year.

They have to be mature tomatoes to make viable seed.

Well that sucks but O.K. At least I still have one big healthy tomato and a few more starting out.

If it's just a small spot, cut it off and eat the tomato.

It is NOT a disease, it will not harm you, and it will not spread to other tomatoes.

Growing Tomatoes FAQ:

What is blossom-end rot? How can I prevent it?

Blossom-end rot is a disorder of tomato, squash, pepper, and all other fruiting vegetables. You notice that a dry sunken decay has developed on the blossom end (opposite the stem) of many fruit, especially the first fruit of the season. This is not a pest, parasite or disease process but is a physiological problem caused by a low level of calcium in the fruit itself.

Symptoms

BER, or blossom-end rot usually begins as a small "water-soaked looking" area at the blossom end of the fruit while still green. As the lesion develops, it enlarges, becomes sunken and turns tan to dark brown to black and leathery. In severe cases, it may completely cover the lower half of the fruit, becoming flat or concave, often resulting in complete destruction of the infected fruit.

Cause

Calcium is required in relatively large concentrations for normal cell growth. When a rapidly growing fruit is deprived of calcium, the tissues break down, leaving the characteristic lesion at the blossom end. Blossom-end rot develops when the fruit's demand for calcium exceeds the supply in the soil. This may result from low calcium levels in the soil, drought stress, excessive soil moisture, and/or fluctuations due to rain or overwatering . These conditions reduce the uptake and movement of calcium into the plant, or rapid, vegetative growth due to excessive nitrogen fertilization.

Management

Adequate preparation of the garden bed prior to planting is the key to preventing BER. Insure adequately draining soil in the bed by adding needed ammendments, maintain the soil pH around 6.5 - a pH out of this range limits the uptake of calcium. Lime (unless the soil is already alkaline), composted manures or bone meal will supply calcium but take time to work so must be applied prior to planting. Excess ammonial types of nitrogen in the soil can reduce calcium uptake as can a depleted level of phosphorus. After planting, avoid deep cultivation that can damage the plant roots, use mulch to help stabilize soil moisture levels and help avoid drought stress, avoid overwatering as plants generally need about one inch of moisture per week from rain or irrigation for proper growth and development.

Once the problem develops, quick fixes are difficult. Stabilize the moisture level as much as possible, feeding with manure or compost tea is recommended by many, foliar applications of calcium are of questionable value according to research because of poor absorption and movement to fruit where it is needed but many have reported that foliar application of magnesium (epsom salts) can effect added calcium uptake. Other various suggestions consist of powdered milk, crushed egg shells tea, bone meal tea, Tums tablets, etc. but prevention is the key. Some recommend removing affected fruit from to reduce stress in the plant.

BER should not be confused with fruit abortion or inadequate pollination although the symptoms may appear similar. The onset of BER occurs only after the fruit is well on it's way to development while insufficient pollination problems terminate the fruit while still quite small.

Here is a link that might be useful: BER FAQ

I plan to sprinkle water containing calcium nitrate on my tomato and pepper plants this year using a watering can. Sort of foliar feeding, but I guess technically not since I am not spraying the underside of the leaves. Whatever drips off the plants will then be available to the roots. I will let you know how this works out in about a month or so.

What Dave posted from BER FAQs is correct, but in the nutshell, it most often happens to first tomatoes not because there is a lack of calcium in your dirt, but because the root system is not large/deep enough yet to support both fast growing of plants and fast growing of fruits.

Most cases of BER go away on its own... also a reason why there are so many reported miracle BER cures out there.

I agree. In my experience, BER happens to a few of the first fruits, then goes away whether treated or not. Be patient.

Jim

Last year I lost 4 plants worth of tomatoes to BER. I was sooooooooo bummed. I got a maybe handful of extremely small (my romas were the size of grapes) tomatoes.