Moult occurs in their second autumn/winter, so she will be approaching her first proper moult in 2-3 months. Some start as early as July or August whilst others wait until the depths of winter to lose their feathers, silly birds! Usually the first moult will be mostly a neck moult. Some chickens lose their feathers almost overnight which can be pretty scary for someone who hasn't witnessed a moult and seen a bald chicken, others you can hardly tell that they are moulting other than the odd feather under the roost each night.
My finances do not run to vets for my chickens either but just giving you the options that are available.
Do you have a photo of the poop with the egg in it? Was it yolk or albumen? It may be that the egg broke inside her which is not good.... even a shell less egg.... because the contents can then harbour bacteria and allow infection to set in. I'm not one for antibiotic use unless absolutely essential but you might want to make preparations/enquiries about how/where to get some in case she takes a turn for the worse. Keep a close eye out for any abdominal swelling between the legs or around the vent. The fact that she is not laying is worrying and may indicate that an infection has already set in. Salpingitis (infection of the oviduct) causes the tract to become impacted with "lash egg" material, a fleshy mixture of egg and pus. Do a google search for lash egg images because you need to keep an eye out for her passing one. These build up in the oviduct and can become the size of a grapefruit and eventually it constricts the gut and the bird dies of toxic shock in a similar way to egg binding, because they cannot pass waste. Or because the oviduct is blocked and backed up, the yolks start dropping into the abdominal cavity (internal laying) where they can become infected and cause Egg Yolk Peritonitis or fluid builds up because of them (ascites). Sadly all of these ailments usually prove fatal, but if you can catch Salpingitis in the early stages it may respond to antibiotics.
