Stuttering (and few other speech related abnormalities) are often perceived as a sign of low confidence levels. People who do not normally stutter in their every day life may stutter during interviews or when talking to new people or during presentations etc when they are low on self confidence.
Rejection for certain roles may indeed happen on such aspects when being able to speak clearly is a requirement for the role.
"Equal opportunity" employer means that the employer gives equal opportunity to every one fully capable of fulling a role regardless of race, color, region, religion, physical disabilities etc. It does not mean that people not capable of doing full justice to the requirements is also considered for the role(though the company would still grant them the opportunity to try). It all depends on the role and circumstances.
Just to explain things, lets take this scenario
Role1: Excellent in Skill-A and Excellent communication skills
Role2: Excellent in Skill-B, requires a lot of physical exertion like walking
There are 3 applicants
1. Candidate-1 : Excellent in Skill-A, average in Skill-B, Average Communication skills.
2. Candidate-2 (Physically challenged): Excellent in Skill-A and Skill-B, Excellent communication skills
3. Candidate-3 (Speech Disorder): Excellent in Skill-A and Skill-B
For Role1, Candidate-1 and Candidate-2 have equal opportunity of getting the job and Cadidate-2 is likely to be chosen due to his better communication skills despite being physically challenged. Candidate-3 may also apply for the role, but he does not have the same chance of landing the job since his speech disorder keeps him from fulfilling the requirement of the role.
For Role2, Candidate-1 and Candidate-3 have equal opportunity, but it Candidate-3 is likely to be chosen due him being better at the the core skill. Candidate-2 is also good at the core skill, but due to the other requirement he may not be able to fulfill the role which is why he does not have the same chance of getting the job.