Any student may request an independent review of his or her disciplinary case.

 “Any” is singular. In this example, we must choose a singular pronoun that applies to a person ("student") rather than a thing. Our choices are “his” or “her.” The sentence does not indicate that the students are only one gender, so we must use “his or her.”

Neither of the men would admit his error.

“Neither” is singular because it divides people or things into two individual parts. In this example, “neither” separates one man from the other and the sentence focuses on the behavior of each man individually (singularly). Therefore, the sentence requires a singular pronoun.

Both the marketing team and the development team asked how much funding they would receive for the project.

“Both” is plural because it joins the two subjects ("marketing team" and "development team").

The company evaluated the customer base it had developed.

If you refer to a company as a whole, it is a single unit, so the noun is singular. (Note: If you refer to the people who make up the company -- for example, “the company employees,” “the members of management,” etc. -- the noun will be plural.)

My co-worker, Rita, began to explain her absence.

"Rita" is a singular noun and so requires a singular pronoun.

Ms Arias, along with her associates, presented her findings.

This sentence contains a dependent clause set off by commas (“along with her associates”). Because the information set off by commas is not critical to the sentence, ignore those words and focus on "Ms Arias" — a singular noun.

The shift leader and the employee discussed their disagreement.

The pronoun is plural because the subject of the sentence is made up of two nouns, the shift "leader" and the "employee."

Several of the inventory clerks still submit their reports in writing.

“Several” is plural because it refers to more than one of something.

Every employee is responsible for securing his or her belongings in the locker room.

“Every” is singular because it focuses on the individual parts of a collection or individual members of a group. 

In this example, we must choose singular pronouns that apply to a person ("employee") rather than a thing. Our choices are “his” or “her.” The sentence does not indicate that the employees are only one gender, so we must use “his or her.”