Thanks for the reply, Joyfull. Sorry about taking so long to get back to you - we have interminable problems with our Internet connection.
The hens were given to us about 18 months ago. We were told they are Rhode Island Reds. Certainly they look like the images of RIRs I've seen on the Web. As far as I know, they're just over three years old now.
They have no access to oyster shells. It's impossible to buy calcium supplements of any kind in this part of the world. I've tried everywhere you could possibly imagine - vets, agricultural coops, feed outlets, hen suppliers and hardware stores.
The feed they have is supposed to contain everything they need, including calcium. Over the winter, I supplemented it with rice or pasta, a small amount of fatty, protein-rich food (usually chopped almonds but sometimes meat, fish or grated cheese) and some cooked vegetables. They each laid an egg just about every day, despite the extremely cold weather (most nights around minus 12ºC, occasionally lower). Since the problem occurred, I've tried giving them just feed, thinking that maybe they were eating my goodies and not getting the calcium they needed, but it has made no difference. Their wattles and combs have faded slightly since I changed them over just to feed.
The things I've found that I thought were lashes are rubbery looking things that look like layers of shell with pink ham-like stuff and a foul-smelling watery liquid inside. The hen with the problem lays one of these every ten days or so, along with the occasional egg, without a shell or with a shell so thin it breaks when the egg is laid. Occasionally she lays more than one egg a day. They often pop out as she is walking about.
These problems apply to just one of our hens. The other one is still laying a decent egg just about every day.
Any and all advice very gratefully received. Around here, people would just 'dispose' of her and get another hen. The vets are obviously not interested in her.