I have two Female Cuckoo Lemon Orpingtons and one Bresse Rooster. I was wondering what color would the chicks be
Bresse Farms offer Blue, Black, and White Bresse. I'll include a link. I assume yours is White because you did not correct that assumption. Looks like Paradise only offers White Bresse.
Bresse day old chicks, starter chickens and juvenile chickens - BRESSE FARMS
I don't know genetically what makes a Lemon chicken. I tried searching online and got different possibilities. There may be more so I don't know what to use.
Th Cuckoo affect is the barring gene when it is paired up with the fast-feathering gene like a Dominique or Cuckoo Marans. The feathers grow fast so the barring affects are scattered. If that barring gene pairs up with the slow-feathering gene you get crisp barring like the Barred Rock. This gene is a sex linked gene so a not-barred male over a barred (Cuckoo) female will give you a black sex link. They don't have to be black to be a black sex link, it's the barring that counts. So any girls that hatch from your cross will not be barred and any boys will be barred.
But I saw one post that said Lemon Cuckoo is based on Mottling, not Barring. That is a totally different gene and would disappear the next generation. No Sex Links.
There are two ways to make a solid white chicken, either the Recessive White genes or the Dominant White genes. If your white boy is based on Recessive White that will turn any feather that would normally be another color white so you don't know what might be hiding under it. Since it is recessive unless it pairs up it will have no effect the next generation since your hen is not white.
Dominant White modifies what would normally be a black feather white. It doesn't affect other colors. That means if a chicken based on Dominant White is solid white it has to be based on solid black underneath. Black is pretty dominant over most colors. If he is based on Dominant White as mentioned above, then theoretically you should get all solid white offspring.
But that's theory. Some genetics modify black feathers to be different colors. Your lemon hens may contain some of those. Buff chickens often do. I don't know what would happen if Dominant White clashed with some of those. I'm not just talking about leakage, you might get some interesting patterns on those chicks.
If the male offspring are barred and you get solid white chicks you will not be able to see the barring, it will not show on solid white feathers. If you get an off-white chick or some other color feathers you should be able to see barring if it is there. For all I know your White Bresse may have barring. Breeders sometimes add barring to white chickens because it makes them look whiter.
and would I get three different body types?
You might get more than that, at least to a certain degree. There are a lot of different genes that control the various aspect of body type, some dominant and some recessive. Not all chickens of a certain breed from the same source all look identical in body type, which means they can have different combinations of those genetics. When you breed them you can get different results. Some of these differences may be subtle, some can be more noticeable. One of the big differences I regularly see is that they mature at different rates. That can be important in meat birds.
In general I'd expect the offspring's body type to be somewhere in between the parents in the first generation when you cross breeds. If you breed the offspring you can get a lot more variations. Same thing is true about color. I'd expect the first generation chicks to be similarly colored, whatever that turns out to be, but if you cross the offspring you can get a real rainbow.
I went through one of the Bresse threads and one of the members crossed a Bresse with an Orpington and got white chicks
Do you know where that person got his White Bresse? If it was from Bresse Farms, where you got yours, it is pretty sure it is Dominant White and your chicks should all be white.
I went through all this to say I don't know what the chicks will look like. It mainly boils down to what genetics make that white boy white. If it is Dominant White, the offspring should be pretty much white the first generation. If it is Recessive White all bets are off.
I know it will be a while before you hatch any but I'd like to know what you do get. Could you please post photos when they feather out.