Step 1: Concept
The pronouns "either," "neither," "each," and "everyone" are singular and must be followed by a singular verb.
Step 2: Meaning
"Either of the students" refers to one student or the other, but not both at once.
Step 3: Analysis
Although "students" is plural, the subject is "Either," which requires the singular verb "is".
Step 4: Conclusion
The correct verb is "is eligible".
Final Answer: (B)