One major reason not to do this is: what if she accepts the offer, despite your warnings?
You are now in a very awkward position: your junior is now in a position to blackmail you. This is especially important as you are considering jumping ship: your reputation capital gained at your current firm is at its most valuable. (In fact, you are vulnerable to blackmail or the consequences of inadvertent disclosure whether or not she accepts the offer.)
Regardless of whether the moral thing to do is telling the excellent former intern that your company is toxic and destined for the corporate dustbin, you are now putting your own career in jeopardy.
A missed opportunity is a relatively minor thing early on in your career. Later on, being seen to act against your employer's interests and in a matter that is entirely work related... well, that's a line on your CV that you don't want.
Regarding the decision itself and its potential consequences, let's look at its moral, legal and financial aspects, examining each separately.
Morally - telling her the lay of the land is absolutely the correct thing to do. (Companies' personalities are a legal fiction and you owe them no moral loyalty - pay them no heed when considering right and wrong.) However, no one should presume to tell you that you must put the interests of others before your own.
Legally - Presuming a common-law jurisdiction, there's multiple potential heads of claim your company might pursue against you, if they were to find out about your thwarting their attempt to recruit the intern. The most relevant tort is tortious interference with business relations. There might be a case to answer for defamation or possibly malicious falsehood, depending on exactly what is said. (Truth is always a defence in defamation actions, but - roughly - if you say something that would make someone think less of another, they can establish a prima facie case.)
Actions in contract are also very possible, even if your employment contract doesn't appear to cover this explicitly: good faith and fair dealing is an implied term in all contracts (in the US, at least). The duplicity involved in paying lip service to your duties to your employer while secretly undermining them demonstrates clear bad faith on your part, and would be a breach of contract. (Some other answers seem to suggest that your duty to your employer stops when you clock off: this isn't true.)
If all this seems unfair to you, consider if you interfered with a potential client for your firm similarly, submitting a tender to satisfy your boss while privately advising the new client not to accept it. From a legal perspective, this situation is pretty much the same, although in this example we don't have the emotive factor of a deserving young person potentially being harmed by your inaction, so it is perhaps easier to see why the law works the way it does. (Regardless, the legal system is largely indifferent to whether we might think it is fair or not.)
In addition, regardless of whether or not you are doing the company damage in legal terms, if your current company were to sue you on any grounds, it wouldn't be hard for a good attorney to find some pretext to get this situation into evidence to (correctly) portray you as disloyal. (This is particularly bad if you are in a jurisdiction where a jury is the trier of fact in civil cases.)
Now, all of this is contingent on the company finding out about you telling the intern and being able to prove it. This is, in itself, unlikely, if you are careful, so while nothing will probably come of this, legally, you're doing yourself no favours at all by telling the intern how things are - you are creating potential liability on several fronts, as well as giving your firm an excellent reason to fire you.
Financially, there's no upside. Only potential downside. You're risking getting fired and getting sued, as well as your future employability, and, grim as it is to say, you are gaining nothing quantifiable - nothing that will put bread on the table - by telling the intern how things are.
No attorney would tell you it was in your best interest to do this. Probably, you'd get away with it, and nothing would happen, other than you would get a warm glow from having gained the gratitude of the former intern. However, if this whole thing went wrong, it would probably go wrong very badly for you.
To repeat: only downside.
By telling the intern how things are you aren't just being honest: you're asking her for her confidence, and that is by no means a given. A particularly mercenary and ambitious personality might even smell an opportunity in your indiscretion.
I can completely understand how someone might nevertheless want to go ahead and warn the intern despite this advice; however, doing so is not the sensible and considered thing to do. It's your life you're talking about, more than hers, and prudence and discretion should be your watchwords.
If this does go wrong for you, it will come from either lack of caution on your part or from the indiscretion - deliberate or otherwise - of the former intern herself regarding your actions here (perhaps more likely if she takes the job, but the legal consequences are such that you are vulnerable either way). So, if you do choose to tell her, consider putting as much distance as you can between yourself and the advice, to minimise evidence and to ensure that there is nothing that can be traced back to you or that otherwise identifies you. An anonymous note, for instance, would probably be taken much less seriously by the former intern than advice not to accept coming directly from you, but it would be rather less likely to come back and bite you as well.