It's not quite like a mule, since it isn't a hybrid, but a tetragametic chimera (a form of congenital chimerism); 'tetra' meaning 4, 'gametic' meaning gametes (sperm and ova) and 'chimera' meaning a single organism composed of two different populations of genetically distinct cells that came from two different zygotes. Four gametes come together (2 sperm and 2 eggs; (first the each sperm fertilize a separate egg, then the eggs fuse together), so it would kind of look like reverse meiosis) to create a completely unique budgie.
Half siders are like the opposite of identical twins, instead of a single zygote splitting into two identical zygotes, it's two different zygotes fusing into one! Just to be clear, a zygote is the initial cell formed when two gametes join.
It happens purely by accident and is not genetic. In birds it obviously occurs in the zygote or blastocyst stage of development before the egg shell forms around the chick. (There are five main stages of development: zygote, morula, blastocyst, embryo, and fetus). It generally occurs when the female has released two eggs from her ovary at the same time by accident and they fuse for unknown reasons after each is separately fertilized by different sperm.
Because the half sider is the combination of two baby budgies fused together, it essentially has two different sets of DNA, so the chick can come to possess organs and body parts that have different sets of chromosomes. For example, in a visual half sider, pigment on one side/part of the body is controlled by one set of chromosomes while pigment on the other side/part of the body is controlled by the other set of chromosomes from the other zygote, which is why half siders are generally two colours (of course they could also be the result of a fusion of two green, (for example), budgie zygotes which would result in a normal-looking green budgie even though it's really a half sider). If the zygotes had not fused, you would have gotten two healthy, perfectly normal budgies (one blue one and one green one). Half siders could also be half albino, half blue or half lutino, half green! It all depends on the parents.
So to answer my own question, no, you can't get a classic half sider from a blue parent and a green parent if the green parent is homozygous green; it would have to be split for blue in order to get a blue and green half sider. If it were homozygous green no blue offspring would ever arise so any zygotes that fused would always ever be green zygotes, no blue ones.
When it comes to breeding a half sider with another budgie it all depends on if the reproductive tract of the half sider was formed from one set of chromosomes or from both sets. If from one set, the half sider can create either Bb gametes or bb gametes if it's a standard blue and green half sider, giving you a normal chance for offspring. If from both sets, it can create Bb and bb gametes, giving you 3x the chance of getting blue offspring from a standard blue and green half sider. But if the gender of one zygote was female and the gender of the other was male when the half sider was formed, there's a chance your half sider budgie is infertile due to a number of reasons including malformed genitalia and hermaphroditism. Hermaphrodite half siders may have two colours to their cere, one side blue, the other side brown. Even fully male or fully female half-siders are usually sterile anyways.
I hope I helped!