How does one's gender affect the Assignment one receives in The Giver?

Gender does not affect most assignments, except for the Birthmothers.

Jonas’s community is pretty egalitarian, including gender.  The community does not like to draw distinctions between citizens.  This is all part of Sameness.  This is why there is really no reference to gender when it comes to assignments.  The only assignment that has to go to a female is Birthmother, because boys are physically not able to give birth.

After the Birthmothers have given birth the requisite amount of times, they are not distinguished from other Laborers.

Three years," Mother told her firmly. "Three births and that's all.  After that they are Laborers for the rest of their adult lives, until the day that they enter the House of the Old. Is that what you want, Lily? Three lazy years, and then hard physical labor until you are old?" (Ch. 3)

The Birthmothers do not raise the babies, or even see them.  It is a temporary assignment at best.  They are nothing more than an incubator for the infants, test tube babies created and genetically modified in a lab.

We know that gender does not affect assignments because jobs like caretakers of the infants, known as Nurturers, go to men as well as women.  Jonas’s father is a nurturer, and he is obviously a man.

Jonas's father's title was Nurturer. He and the other Nurturers were responsible for all the physical and emotional needs of every newchild during its earliest life. It was a very important job, Jonas knew, but it wasn't one that interested him much. (Ch. 1)

We know that most other titles go to either men or women, including very important ones.  The Chief Elder is a woman, for example.  Jonas’s mother also holds “a prominent position at the Department of Justice.”

The job with the most honor in the community, Receiver of Memory, obviously goes to either male or female.  The current Receiver is a male, and Jonas is a male.  However, the child selected before Jonas was a female, Rosemary.  The only other child who is known to have the capacity is also a female, Katharine.

The only time the community seems to care about gender is in creating family units.  The rules are very specific about which genders the family can have.

Two children--one male, one female--to each family unit. It was written very clearly in the rules. (Ch. 1)

The reason for this gender assignment is not clear, except that having a rule requiring one child of each gender and limiting the family unit to two children keeps every family the same.  That is likely the only purpose, and it has nothing to do with differences between the genders.