I do not think there are any legal implications for the company. Essentially, on a working visa, you are on the lowest rung of the ladder.

The ironic fact is that this country is created by immigrants, and is still, probably one of the most "open" countries in terms of hiring immigrants, but employment is a privilege, not a right

I am a bit confused then when it comes to companies using the term Equal Opportunity Employer. Wouldnt it essentially be 'discrimination on the basis of national origin' if an opportunity if forfeited on the basis of visa requirements?

no, because no one choses to be a man/woman, a disablility, or a particular race.  working in a country in which you are not a citizen is not a "right"

Citizen, even though you pose an interesting question, your argument is flawed. Here's why;

1. If you are on a tourist visa, the essential purpose is TOURISM

2. if you are on a student visa, the purpose is STUDIES

3. Now, the whole premise of a  work-visa being granted is that you have employment. Without employment, you are not authorized to work anyways. What I am trying to say is that the visa follows the work, not the other way around. What they mean is that they are not ready to sponsor anyone seeking a work visa. And that sucks.

@sameolddoctor : This is of course assuming someone already has a work visa and is legally authorized to work in the US. If not, then the company can decide if they want to invest in an individual. 

My question, though, is mainly about the whole issue of Equal Opportunity Employer. Does it mean that the guidelines of Equal Opportunity Employment applies only to citizens of the US? (In which case discrimination person based on national origin would be read as discrimination against a US citizen of foreign origin)

"EOE prohibits employment discrimination based on race, color, religion, sex, or national origin"

Would this apply to someone who is fully qualified for a job but is not given an opportunity only on the basis of

1. Having a visa already?

2. Needing a visa to begin working?

3. Not being a citizen?

the EEOC's faq page says that discriminating based on citizenship may violoate the law. 

My question, though, is mainly about the whole issue of Equal Opportunity Employer. Does it mean that the guidelines of Equal Opportunity Employment applies only to citizens of the US

NO, a work permit means that it is legal for you to work, and as long as you are here legally ,all people in the US have equal protection under the law.  It violates the commerce clause to discriminate based on a visa for most jobs.  certain jobs like police may be exempt from this. 

I've even heard that firms that have accepted government stimulus money are prohibited from hiring people on visas, as the purpose of the stimulus is to create American jobs. Equal Opportunity means equal for all Americans, I guess :)

@anarchytecture, an architecture firm getting a bailout!? I think that's highly unlikely! Haha

I am quite certain that firms refusing employment to someone based on not having to sponsor them is completely legal.

Citizen, if someone has a work visa and they go to work for a new company, the new company HAS TO re-sponsor them.

Yes, opportunity should be based on factors like skill, education and worthiness to a company, but it will be too much for fuckheads like the DOL to understand these things!