Isolating calcium acetate from water with electrolysis
How to get rid of water to collect calcium acetate? 

Evaporate it. 

How to do it in an environmentally responsible way?


Electricity from the outlet cost you money and probably used some
combination of coal, natural, gas, and or nuclear to produce it. 
Bunsen burner, but it likely uses a fuel that was taken from the ground, made from a crop that that competes against food production for land.
Solar cells to produce electricity. It uses renewable energy but
is relatively expensive and electricity is a high value energy
compared to solar thermal. Using commercial grade solar cells only
converts about 17% of the energy in sunlight into electricity that is then converted into heat to evaporate the water.
Solar thermal is cheap and easy to make with household items like an aluminum or soup can and black spray paint. Further more, it has the potential to be more than twice as efficient as solar cells at providing energy you need to evaporate water.


Electrolysis:

If you insist on electrolysis for experimental reasons, calcium acetate is a salt of acetic acid and calcium from the calcium carbonate in the egg shells. As a salt, like all salts, calcium acetate will dissolve in water forming ions that will conduct electricity. So, there is no need for any electrolyte other than water and calcium acetate. 

Furthermore, your end products will be something other than calcium acetate and would depend in part on your anode and cathode, temperature, and applied voltage.