This does appear to be a hernia; let me get some pics of my girl tonight that has one. The only thing I question with your girl is that you say it feels squishy/soft; my girl's is actually fairly hard feeling, not like a tumor/mass, just very, very firm, and my girl's is larger, too. My girl laid all last year without issue, and when I needed to take another bird to my avian vet, I took this girl, too, so I could find out what the mass was; the vet inserted a needle and tried to get fluid out to test, but no fluid came out in the needle. She then palpated the mass, and said that she had a hernia, which is actually fairly common (although this is the one & only I've had in my flocks). Vet indicated that it could be caused from early laying...folks that push their birds to lay early by automatically feeding layer feed at a certain time (typically 18 weeks) rather than just letting them start to lay on "their" schedule before starting layer feed. I think my girl began laying fairly early (she's a Black Jersey Giant crossed to a New Hamp. Red hen), like around 22 weeks, and that's when she was put on layer feed. the vet DID say this is just one reason it could happen.
There are many types of fluid buildups that can occur, too - I found this quickly by Googling:
I had two hens fall ill with this same weird ailment a few months back--distended abdomens (which felt like fluid, not solid tissue, to the touch), runny poops and dirty vent feathers, drinking lots of water. I took them to my vet and it kinda mystified him, as well. He finally gave me some Lasix (a diuretic) and Baytril (antibiotic) to give them, he also gave them a Lasix & Baytril injection as well. They pooped water like mad as the Lasix worked to reduce the fluid build up. They SLOWLY improved and while the large abdomen did not fully go away, they were perking around and eating normally. Both were older hens.
I had another hen who developed the same symptoms about three weeks after the first two, so we took her to the vet and got the same treatment for her. She started to improve greatly, and then suddenly died.
She was only 2 years old and had been as healthy as a horse.
All three hens had been robustly healthy beforehand. The only thing out of the ordinary that had happened just before they got ill, was a strange outbreak of tiny black & orange caterpillars in one of our trees, which the chickens refused to eat--except I think that these same three hens had each tasted or swallowed one. When chickens won't eat a bug, it MUST be something awful.
Poison caterpillars? Best I could narrow them down to was Gypsy Moth caterpillars, but it was not an exact match.
The vet still seems to think it was some kind of bacterial infection. If it was, it was a resistant strain, because it was VERY tough to kill! The fluid build up and excessive thirst pointed to liver/kidney failure, which is very often fatal. My vet didn't think the first tow hens would live.
If it's simply a hernia, she will likely continue laying and have no issues; my vet told me the biggest issue is they get dirty due to the swollen part sticking out (my girl didn't seem to get dirty - her bulge was down low, almost between her legs). Vet also that that eventaully, she'd likely become an internal layer and that's what she'd die from. Sigh...BEST of luck with your girl! If she's still laying & you've got a roo, I'd strongly consider hatching many eggs now, while she's still laying.
