I'm pretty interested in topics like this so I've been thinking about it a lot. I've seen a lot of lists like yours online that demonstrate a certain pattern to the prize order for the blue and pink boxes. The gold box is a separate issue I'll come back to at the end.
Supposedly, you win gold eggs in the blue and pink box 45% of the time, fences at 20%, towel/tree each at 12.5% if pink is your common egg color 15% if it is your uncommon color, 12.5% for each of the egg pile/pond if blue is your common egg color 15% if it is your uncommon color, 5% for the Lizards/Blocko Store (pink common) 2/3% (pink uncommon), 5% for Egg Council Guy/Johnny Fiestas (blue common) and 2/3% (blue uncommon).
It's very clear you win 2 gold egg prizes for every 5 boxes with a 3rd gold egg prize the 4th time you go through a set of 5 such that you win 9 gold eggs out of every 20 boxes (45%). I think 45% isn't so much a random chance to win the gold eggs as it is a description of how many gold eggs everyone would actually win if they opened 100 boxes. It's definitely scripted. If it were truly random, someone would have won gold eggs 3 times in a row. That should happen 9% of the time under truly random conditions given those odds. I actually think it makes sense that they would script the gold egg prizes. If they don't, a certain percentage of the players will win gold eggs 10 times in a row and another similar percentage would win 0 gold eggs in 10 boxes. Neither of these are desired results. The people who got lucky and keep winning gold eggs repeatedly are going to win every gold box prize pretty easily and never spend money on eggs. The people who win no gold eggs are going to quit and never spend money on eggs. By scripting out the gold eggs, they ensure everyone has a similar amount of chances at the gold box, and ensures that people keep coming back for more torture.
The other thing I noticed about the prize lists is that non-gold egg prizes never repeat twice in a row. Ignoring the rarer prizes for a moment, there are only 3 non gold-egg prizes per box. Thus if a prize never repeats, then lining up your prize lists like this is going to look very similar. There pretty much is only 2 prizes that can come after a fence if you recently won a bunch of gold eggs so everyone's prize list looks similar no matter what.
Since they clearly scripted the gold egg prizes, I think they also scripted the other prizes in the blue and pink box although due to non-repeating prizes it's somewhat harder to tell. They could be taking a crappy random number generator and rejecting values that don't meet a certain criterion until it comes up with a prize it allows you to win on that opened box. If it's scripted, then that actually helps you know that there are exactly 5 e.g. Johnny Fiesta prizes in the first 100 boxes and means that your odds of the next box containing that actually are going up albeit slowly. I suspect that everyone would win their common box prizes well before the 80th opened box and at least 1 of the uncommon box prizes before the 80th box for this reason.
The algorithm probably works like this (this need not be the only way to do it)
Check what number box this is. If 80/81 award specific scripted prize.
Check prizes won, if gold egg prize count percentage falls too low, award gold egg prize.
Check last non-gold egg prize won, if randomly chosen prize is the same, award different prize.
The Gold box is a different animal. I've seen reports of people not winning one of the topiaries in 80 opened gold boxes. The gold box odds are: Bobbins 2%, Hugs 3%, Sean 5%, Bunny 12%, Dictatorship 16%, Gate 17%, Kang 7%, Kodos 7%, Fence 31%. If the odds were closer to random than the blue/pink boxes, then 3/1000 people would not get one of the topiaries in 80 opened gold boxes. That is often enough that you would probably hear about it. These people would be likely to complain so in an online forum you're likely to hear from them; it's a sample selection effect. It's also worth noting that I won Hugs Bunny and whatever he was replaced by 7 times in 50 gold boxes. If they had scripted it like the blue/pink box, I wouldn't have won more than 3. Meanwhile, I didn't win Father Sean until his scripted box at the 50th opened gold box. At 5% odds, you have a 92% chance of winning him in 50 opened boxes. The only thing I see out of the gold box is that you can't win the same prize twice in a row (unless one of the boxes hits a scripted box, e.g. Banana Dictatorship on gold box 15, and then winning the 250 eggs on the next box as well; I don't think the algorithm counts the scripted boxes). Since the gold box is probably closer to random, getting Sharry Bobbins is actually harder. True 2% random odds will come up less frequently than 2% for many people so getting Sharry Bobbins will require getting to the scripted 80th gold box for probably around 50% of players (I was one of the lucky ones, getting her on gold box 25). Of course, no computer generated random number is truly random but that is a different topic.
