Everything is negotiable. As a previous poster stated, saying something is non-negotiable is just a negotiating tactic.
To all the previous posters who stated you have no leverage, I would say that they are wrong. Even entry-level hires have some leverage, but not a lot.
Think about it from a company's perspective. The company has gone through this lengthy process of reviewing resumes, possibly doing phone screens, doing in-person interviews, and then choosing a candidate and making an offer. That is a lot of time they have invested in the selected candidate. If the selected candidate counter-offers and if the counter-offer isn't outrageous, it might make sense for the company to agree to the counter rather than rescind the offer. If they rescind the offer, they'll either have to now make an offer to their second choice (usually not a great option unless 2nd choice was a close second) or go through the whole process again (additional time and expense). Most likely they will just say "no".
You actually have more leverage than even an entry-level hire because you have worked for this company as an intern. This company already has time and money invested in you in the form of training, knowledge of processes, etc. This is time and money they will have to invest again in a straight entry-level new hire.
Before a big decision, I use this little test: what is the best possible outcome, what is the most likely possible outcome, what is the worst possible outcome. So for me, if I can live with the worst possible outcome, then I go ahead with my plan. Sometimes you have to weigh the answers to the three questions and decide. Maybe you're not willing to live with the worst possible outcome but that outcome occurring is remote and the best possible outcome is amazing and well worth the risk. Then you must decide for yourself.
In this particular situation, I would say your best possible outcome would be for them to agree and meet your demands. Your most likely outcome would be for them to say "no". Your worst outcome is for them to rescind the offer.
If you can live with the rescinding of the offer, then try again. You have more leverage than you think. However, the company may not agree; it is your job to make them see the time and effort they have already expended on you.
If you really need the job, then I would say it isn't worth it. It is difficult to negotiate if you are unwilling to walk away.