Illustrator :: Saving A PNG For BOTH Screen And Print

I have a client who is requesting that their "image" be saved as a PNG at "715px X 915px at 72 dpi", and have it print with the text crisp and clear. I have tried to convince them to use a PDF instead to ensure that the print quality would be where they want it to be however they reject that idea.
 
1. They are requesting 72 dpi not ppi. There is a difference, one is screen one is print. You cant have both in my opinion. I have been using ppi, since that is the setting that illustrator and photoshop allows.
 
2. You also should not save anything for print at 72 ppi. That will not produce the best quality image regardless.
 
3. I have attempted several different ways to save this.
 
- I have saved it in illustrator as a png and mulitple resolutions.
- I have used my illustrator png and put in Photoshop and re saved it, after re sizing it. The dimensions in illustrator never seem to stay accurate.
- i have attempted to save it as a PDF and convert to PNG is acrobat.
 
None of these work. There is no issue getting it to look nice on screen, its the printing part that I am having an issue with. When I save it at anything higher then 72 it increases the dimensions. If I create it at anything higher then 72, the dimensions increase as well. I have not been able to keep it at 715px x 915 px and have it print clearly. They come back and tell me i need to decrease the size, or the text is too blurry.
 
How do I get this PNG to be 715px by 915px at a 72 ppi (or dpi) resolution look good on screen, and print with clear crisp text?