Alinas, not sure if you went ahead and adopted the neighbour's birds.
After reading the thread, I would concur that the neighbour is not feeding them well, which could mean they are not even laying at all, or eating them.
From my experience, pullets may eat the odd egg, and it is usually because they have attempted to lay from a roost or similar; egg falls on ground, pullet often eats it. I have had two of my (now older) girls do that as they first started laying, but grew out of it, and not eaten any eggs since. Fake eggs in the nesting boxes, and putting them there if making noise/behaviour like they are thinking about laying.
As for integration, your chances are likely to be successful. Roughly even numbers, and roughly same ages.
Two weeks ago, I bought 3 pullets (about 6-8 months old) to add to the existing 9yo, 5yo and 3yo. I went for pullets because I needed a faster integration time (adult size, but teenager brain).
How it went initially, is that the old girls freaked out at "big chickens". For about 8 days, each flock was fenced off with half the run each (with the older girls free ranging when the weather was suitable). On day 2, a couple of altercations through the steel mesh fence. At night, I had a small pen constructed in the chicken shed.
On about day 6, the new girls got the whole run while the old girls free ranged. On about day 8, when weather good and older girls free ranging, new girls got to explore the free range area. I had a couple of very minor pecking order pecks, and two moderate altercations - but I was supervising the entire time. I don't have a rooster, and my alpha (9yo) has an injured foot and is caged. She would normally police the new order, but cannot. So I did it. I told off each newbie that attempted a challenge to the older (scaredy-cat) girls. The newbies all knew their names (day 1 was naming) and if I saw a challenge was about to occur, I would call newbie's name followed by strong "no". This worked. That only happened for about 3 days.
We are at day 14, and there are no squabbles (except one attempt by the lowest ranking pullet, who got told "no" yesterday). And although they sort of mostly hang with existing flock groups, there has been a lot of integration and mixing as well, with no dramas. Pretty good result for day 14, even though not fully mingling all the time.
The other key part of that, is that I watched my older girls' confidence level before I began the integration process, given they were nervous about the newbies. Previous integrations with POL even though a similar method of separation, resulted in months of picking on the newbie (big mistake, adding only one).
So I bucked the conventional 'wisdom' of 'not interfering' with pecking order, and on this occasion at least, it not only worked, but worked exceptionally well. I picked this time due to being on holiday, and able to spend a lot of time with the chickens.
I would add though, the newbies still sleep in the separate pen in the shed (9'x9' or 3m x 3m) as I do not think it is time to integrate them in there yet (confined space). I do have the pullets better trained for "bed time", and they go in without too much protest or trying to escape. I will likely integrate them in the shed in a few more days, or the middle of next week, just following my intuition for when the time is right.