From a legal point of view, the commenters below address some of the issues which obtain under these circumstances. But as a non-employed intern you are not going to be able to engage in substantive negotiations with them about the terms of a non-compete/trade secrets clause. You are not a headhunted executive, your negotiation position is different. There is great asymmetry in power in your negotiation. All you can do is sign, if you want this position. If an issue comes up later--if they even remember you six months after you leave, 18 months from now--deal with it then. Or have your new employer deal with it.
Companies are bending over backwards to claim that interns are not employees when in most cases they very much are, except for the payment of wages. It will be interesting to see how the "work for hire" copyright doctrine applies when on the one hand the intern-employing company says, "these guys just hang around, the school sends them, they're not employees" and on the other, "we are entitled to the code, processes and procedures they came up with during their time here."
There was an article a few months ago posted in HN about why it's worth it to take the risk in signing boilerplate. There are very few exceptions--one current dispute is the Hoefler/Frere Jones partnership dispute; one side maintains that an employment agreement that was signed by one of the partners is dispositive of their relationship.
Ultimately, six or twelve strangers (they call them 'jurors') will be dragged in off the street and asked to give their group opinion on the facts.
Your internship is very unlikely to require the intervention of these strangers, so don't worry. Try to learn everything you can.