I think it is likely that the dolomite is more informative about the circumstaces in the burial environment than about the eggshell itself. Dolomite is a common product in sediments. In marine sediments it is usually the result of microbial processes, but in terrestrial environments, it can form as an evaporite. Do your fossils occur in arid environments where evaporites might form? 

I think it is the result of particular taphonomy of the fossils examined, so I agree with the opinion of Gaines.

What size are the dolomite crystals? Microdolomite is often found in materials that were originally high-magnesian calcite. This has been seen in the eyes of some trilobites, and in the echinoderm stereom. Crystal size can give an indication of whether the dolomite originates from the egeshell itself, or has been brought in from an external source during diagenesis. Did you carry out simultaneous EDX analysis?

Also, orientation relationships of different grains might help answer this question. Do you have any inverse pole figure maps that you can show us?

@ Clare- I did not carry out EDX Analysis. Please find the file attached.

Soanl, it is difficult to establish what is going on from the Phase map alone - something at higher magnification might be more useful. I would suggest scanning a small area (one that has been identified from the phase map as containing dolomite) and then creating an inverse pole figure map to determine the crystallographic relationships between calcite and dolmite. I'm not sure what system you are using...I only really have good experience in using OIM software (on an EDAX TLS system) but with this I carried out ChI (Chemistry assisted Indexing) scanning (a post progessing method) with great success.