More elaborate answer
No. The statement about 300 DPI being "best" most likely comes from the fact that this is incidentially their printer's native resolution. If you scan a document at a given DPI setting, say 200, then the digital image contains so-and-so-many (200 to be precise) points per inch of original image scanned. The same is true -- in reverse -- when printing. You have so and so many pixels total, and they represent an image as-seen of a particular size, giving so-and-so-many pixels per inch. The DPI used for printing and scanning can be, and not rarely are, different.
Now, without specifying the DPI, it is hard to tell what size an image should be, except if you explicitly specify the on-paper dimensions. If you had scanned the same document at 300 DPI instead, it should normally still have the same size as-seen, although the digital image is much larger, with many more pixels. Without knowing the DPI, it would appear 50% larger on paper (or on screen). Other than that, the DPI of your images have little meaning.
The person who told you 300 DPI are "best" probably meant to say that your scans generally have a bit too few pixels to give good print results, and cranking up the DPI setting on the scanner would give better results. Apart from higher DPI metadata, you also get more pixels in your digital image (and much larger files).
Does changing the DPI help anything?
After the fact, not really. First, it doesn't actually change anything except metadata. But more importantly, either you do not need this metadata at all, or you absolutely do need it, but then it must be the original value!
In a book, you most probably have your images layouted at a given size (often so they match exactly the body width, or one half that), and the digital image is resized-to-fit and resampled automatically during print.
Or, you may not care too much about the layout and instead want the image to appear at the same size as in the scanned original. In that case, you need the original DPI figure. Meddling with it gives wrong results.
Does changing the DPI and resampling help anything?
Not really, but well, possibly a little. That's what you have been doing (hence the file got much larger). The inherent problem of resampling/rescaling is that you cannot magically generate information that isn't present within the scan in the first place. You see that kind of thing on the various "CSI" TV series where they zoom in on a selfie, revealing readable letters on a number plate some 500 meters away accidentially caught in the image. Reality doesn't work that way. You generally cannot magically generate information that just isn't there, no matter what supercomputers you have.
However, I am saying "possibly a little" because there is the odd chance that Photoshop (which is not limited by realtime constraints) uses a slightly more favorable resampling method than the printer. It might look very slightly better than the upscaling that the printer is doing. Maybe. Possibly. Or maybe not, it really depends.
Does scanning at higher DPI help anything?
Absolutely. Ideally, you scan and print at exactly the same resolution. Anything else, and the printer must scale/resample. Either one is sub-optimal although scaling down is generally not such a big issue. Scaling up means you must magically generate image information out of nowhere, i.e. you must interpolate the available information in some way.
One may thus be inclined to always scan at the highest possible resolution even if this exceeds the print resolution (and to be honest, that is what I am doing because more is always better) but very strictly, from a technical point of view, this approach is wrong.
Downscaling, too, means resampling, and resampling is never 100% trouble-free (unless in trivial cases like exactly 1:2).
Depending on how well the downscaling is implemented, too high DPI could actually give inferior results, in theory you could get all kinds of artefacts (think e.g. Moiré patterns, ringing, whatever).
In practice, I've never seen this being a problem, actual printers seem to do pretty darn good filtering.