I will also be printing this document so it would need to be high-res.
In InDesign the output resolution of any placed image is dependant on its scale—the output resolution is listed as Effective Resolution in the Link Info panel. If you place a 72ppi screen capture, and scale it to 50%, its output resolution will be 144ppi—at 24%, it would be 300ppi.
if your printer insists on "highres" screenshots, because an automatic preflight will flag "lowres" images in your provided print PDFs, you could "upsample" the images by e.g. doubling the pixels in PhotoShop to trick that auto preflight. That would not hurt the contents. Best create the screenshots as PNGs if you have a choice.