This isn't really helpful but the OP's birds appear to be Red Sexlinks. I never have soft egg problems until I had them. Then I got rid of them and the problems went with 'em.
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This sounds like to kind of forced the oyster shell on them...mixing it with the corn get them to gobble some shell along with the corn.Do you have any cracked corn available to them?? I had the same problem! Shell-less eggs! I put out a bowl of strictly cracked corn mixed with oyster shells and two days later we have hard eggs!
Vitamin D3 is the sunshine vitamin...great observation!i figured out our problem i think.... i have an enitire flock of bantams who have no laid since i got them in october(well i got 6 mille fleur eggs months ago) so i told OH to throw them out to free range because i was not letting sit on their behinds if they were not laying eggs. They have been outside for 3 days and we have found 3 eggs...so i think in my case it is a vitamin D problem, the coop sits behind the well house and it get light late and dark early so maybe not enough sun in that location for them.