I think it is important to be accurate about the terms you use to describe symptoms or eggs. To me "rubber eggs" means lash eggs because colour and texture wise, they are similar to rubber. Shell less eggs with just a membrane around them should be described as that and are just normal eggs that have failed to have a shell formed around them for a number of reasons. As someone else mentioned, laying shell less eggs is really hard work for a hen and they will usually appear pretty sick when they are struggling to pass one. If you used the term rubber egg to the vet at Cornell he may well assume you mean a lash egg unless you specifically described it as a normal egg without a shell. A lash egg is a mixture of egg material and solidified pus due to an infection of the oviduct (salpingitis) and antibiotics would be the correct course of action although usually it has progressed too far and the oviduct is impacted with lash egg material by the time the problem is identified and treated. It may be that your first sick girl developed salpingitis and eventually became impacted (the swollen back end) with a mass of lash egg material which eventually became so large that it pressed against the gut and blocked it. The bird is unable to pass waste and their digestive system gets backed up (hence refluxing fluids from her mouth) and they die of toxic poisoning.
At 3 years of age production birds are prone to reproductive issues but it may be related to diet since you have had 2 birds with similar symptoms in a short space of time.... what type of layer feed to you give them.... crumbles, pellets, mash or whole grain mix? I would not mix scratch into their regular feed. It is designed to be scattered on the ground to encourage them to "scratch" for it. That stimulates them physically and mentally which can be beneficial but needs to be balanced against the low protein, high carbohydrate content of the scratch which can cause other issues like obesity and fatty liver haemorrhagic syndrome and laying issues like prolapse and perhaps even internal laying. Scattering it on the ground also means that all birds get more or less an equal share of it, whereas if it is placed in the feeder, dominant birds will hog the feeder and eat as much of it as they can. It also encourages them to bill out their regular feed to find the bits of scratch. The billed out regular feed sometimes gets wasted or encourages rodents that come and clean it up at night. They will not leave any scratch to waste if you scatter it on the ground because it is their preferred food/treat. If they are eating scratch then they will not eat as much of their formulated feed which means they are not taking in the nutrients and trace elements that they need including calcium which is needed to make strong shells. Another possibility is that they are eating lots of leafy greens like spinach/kale which I have been told prevent the uptake of calcium.
Just some things to consider in tackling this problem. I cannot advise on antibiotics because I am in the UK and I personally feel that a vet or doctor should be the only people to prescribe them, due to over/misuse and the rise of resistant bacteria.
At 3 years of age production birds are prone to reproductive issues but it may be related to diet since you have had 2 birds with similar symptoms in a short space of time.... what type of layer feed to you give them.... crumbles, pellets, mash or whole grain mix? I would not mix scratch into their regular feed. It is designed to be scattered on the ground to encourage them to "scratch" for it. That stimulates them physically and mentally which can be beneficial but needs to be balanced against the low protein, high carbohydrate content of the scratch which can cause other issues like obesity and fatty liver haemorrhagic syndrome and laying issues like prolapse and perhaps even internal laying. Scattering it on the ground also means that all birds get more or less an equal share of it, whereas if it is placed in the feeder, dominant birds will hog the feeder and eat as much of it as they can. It also encourages them to bill out their regular feed to find the bits of scratch. The billed out regular feed sometimes gets wasted or encourages rodents that come and clean it up at night. They will not leave any scratch to waste if you scatter it on the ground because it is their preferred food/treat. If they are eating scratch then they will not eat as much of their formulated feed which means they are not taking in the nutrients and trace elements that they need including calcium which is needed to make strong shells. Another possibility is that they are eating lots of leafy greens like spinach/kale which I have been told prevent the uptake of calcium.
Just some things to consider in tackling this problem. I cannot advise on antibiotics because I am in the UK and I personally feel that a vet or doctor should be the only people to prescribe them, due to over/misuse and the rise of resistant bacteria.