I don't think that most of the media is reporting this properly. There are some 9,000,000 Filipinos working overseas and most of them would be in the Middle East. When the truck driver was taken hostage, it reminded Arroyo (and the voters) just how vulnerable these people are, especially because of their country's military participation in the Middle East. By having soldiers fighting in Iraq (although I think most of them were mechanics), the Philippine government was putting the lives of all these workers at risk. Even ignoring the moral problems, these 9,000,000 workers (who send most of their pay back to their Nonnos and Nonnas at home) are the Philippines' largest source of international currency - they're probably even more important to the economy than the local workforce.
Then taking into account the fact that the majority of Filipinos were opposed to the war in the first place, and the fact that groups like the Moro Islamic Liberation Front are just itching for a reason to shoot something, especially after being stirred more than a bit by the recent national and local elections, I think that Arroyo would have to be either very brave or very blind to keep the 51 soldiers in Iraq.
Personally, I find the reasoning behind the "we don't negotiate with hostage takers" line flawed. The idea is that, if the government never gives into hostage takers, then the terrorists will eventually work out that they have nothing to be gained by taking hostages and will give up. But I'd think that the hostage takers are taking a similar line. They'd say that if they keep taking (and killing) hostages, then the occupying powers will eventually work out that they have nothing to be gained by taking such a hard-nosed line and will give up. Nothing changes and all the while innocent workers are being abducted and killed. But that's just my view.
btw, I'm half Filipino (in case you can't tell), just so that you know where my biases lie.