If you get good grades, volunteer at the hospital, play varsity sports, and write for your school paper, you might be eligible for a college scholarship.
If you’re eligible for something, you’re either qualified for it or allowed to do it. You’re eligible for unemployment benefits if you’ve recently been laid off, but not if you've never had a job in the first place. You may not be eligible to enter a sweepstakes if you work for the company giving away the money. Neither your dog nor your friend from Australia is eligible to run for president of the United States, but you might be!