Did you mean to say that your final format for print is JPEG? A PDF which would be a much better choice, as it would not rasterize your type like a JPEG will.
In any case, if you started with images at lower than 300 ppi, there is no magic that can add resolution. You can, of course, scale the photo up, but info that was not there to begin with cannot be created by up sampling.
You can only down sample when you export, there's no option to up sample either on export or from within InDesign. To upsample you'll have to edit the image in a photo editing app.
…the printers have said the whole document is at a 65dpi resolution…
If you start with a 300ppi image and end up with a 65ppi image, one of two things happened. Either you downsampled it to 65ppi, or your size was to small and when increased to full-size, the effective resolution became 65ppi (that would happen if you enlarged a 300ppi image to around 460%). If you export to PDF, like the others suggest, you can leave the resolution of bitmap images unchanged, or downsample to a threshold that you choose, while leaving vector elements (like type and vector shapes) in vector format, where the output device will raster them at the resolution it is set to. But your iPhone screenshots won't get better resolution, if that was your question.