Autosexing is base on the barring gene, which is on the sex chromosome (Z). A female has chromosomes ZW, so she only has one copy of the barring gene. A male has ZZ, so he can have two copies of the barring gene, so he gets a bigger effect (has more white than the females).
Yes, Legbars (blue eggs) are supposed to be autosexing, as are Bielefelders (brown eggs), and also Barred Rocks (brown eggs) and Dominiques (brown eggs). In practice, some are easier to tell than others.
Welsummers are not autosexing.
Welbar would be a specific breed that is autosexing (like a Welsummer, but with barring.)
Sexlinks are a bit different. You can sex them by color at hatch, but it won't work for the next generation. Sexlinks can be barred/not (Black Sexlinks and Blue sexlinks: barred mother, not-barred father produces barred sons and not-barred daughters.) Sexlinks can also be gold/silver (Gold Comets, Red Stars, and many other names. Gold father x silver mother gives silver sons and gold daughters.) They can be chocolate vs. not (chocolate father x black mother gives black sons and chocolate daughters.) They can even use leg color (dark-leg father, light-leg mother gives light-leg sons and dark-leg daughters.)
For that, you could use autosexing or sexlinks, if you don't intend to keep males and breed more chicks.
There are lots of options that lay brown eggs, but Legbars are the main ones that lay blue eggs and I can't immediately think of any that lay white eggs that are autosexing or sexlinks.
Since you've already identified a few hatcheries, I would suggest you look at what breeds they have available, then do a little research on each breed. You'd mostly be looking for autosexing or sexlink, and egg color, and those things are usually mentioned in hatchery descriptions.
