None of the responses so far have really hit the nail on the head.
DPI doesn't work like you think. Cameras record images at a certain pixel size. All DPI specifies is how spread out those pixels are when they're actually printed. For example, 300 DPI means 300 pixels per inch, so the 2560x1920 pixel image from the Coolpix 5700, if set to 300 DPI, would print at 8.533x6.4 inches (2560/300). Whether the image that comes out of the camera set to 72 DPI or 300 DPI is irrelevant as changing that number does not affect the image in any way, only how large the printer reproduces it.
A word of warning: In Photoshop on the Resize Image screen, if you have "resample image" turned on, changing the DPI will have an unintended effect. I'll attempt to explain.
Let's say I have a Coolpix 5700 image. 2960x1920 pixels. Lets say the file that comes out of the camera is set at 72 DPI (I don't actually know, but for the sake of this example...). That would mean print size for the image would be approx. 35x24 inches. Now, let's say you have "resample image" on. If you change the DPI to 300, it will keep the print size at 35x24 inches and up-sample the image to 10667x8000 pixels. You DO NOT want to do that, as such severe up-sampling will destroy image quality/detail/everything. If, on the other hand, you have "resample image" turned off and you change the DPI from 72 to 300, it will leave the pixel size alone and change print size to 8.533x6.4 inches. Again, making this change has not changed your image in ANY way, merely the way the printer reproduces it.
Your employer is right, however, in saying that all printed images must be at 300 DPI. Don't listen to anyone who tells you otherwise because except for on Newsprint, even down to 200 DPI looks like crap. I hope I've made it clear however that all cameras are "capable" of producing 300 DPI images. How big those images are, however, is another matter. To determine whether a camera has enough resolution to meet your needs, divide the length of the sides (in pixels) by 300. The resulting numbers are the dimensions of the image in inches.
I'm not sure if any of that was easy to understand, feel free to email me at firstname.lastname@example.org
if have specific questions or if the entire thing was unclear.