If this is correct "The screenshots I am capturing are at 72 ppi," then "Photoshop and increasing the resolution to 300 ppi."
Are you capturing image from other digital media?
The problem lies in the original image/screen capture. You can either capture a Higher PPI image or work a little harder around the issue but you are restricted by the original PPI. What your really doing is increasing the pixel count per inch.
If a pixel had two different colors in it from the re-size problems arise. Each re-sizing program will have different outcomes due to the math algorithms behind them.
If the problem is captured text.
DON'T BE LAZY!!!! Use InDesign to format the text. That means either type in the text of the screen capture or us an OCR to turn that text-in-image into copy and paste text. I would think you could just copy and past in the first place unless you are scraping a flash video slide show or something and if so way beyond what I can help you with.
Is it just printing issues?
Printers print at 300dpi you will start losing quality because the printer has to guess what colors to put into the larger pixels (the opposite of the above image).
Setting issues: InDesign or PDF
InDesign has a bunch of setting that you can change one is image preview (not sure of actual name) and you have a entire projects setting and an individual image override setting with the right click. This should not affect the final outcome at all print it as a PDF and if the image is blurry it is the way you are trying to use the images.
Making small(1 inch) image bigger(3 inch)?
CAN NOT DO WITH A SCREEN CAPTURE!!!! They are bitmap images not vector you will need a vector image and they will be clear at any size. No answer can not fix. Must remake the entire image from scratch.