What type and size plywood is appropriate to reinforce a box spring for 425 lb user?
Where is the damage in the box spring (middle/side edge/corner) and how much of it needs reinforcing?  If the damage is in the mattress with seemingly good supports all around, just about any plywood half inch or thicker is going to work fine for this purpose.  Since its going to be 100% non-visible, buy the cheap stuff.  If its on the edge or corner, half inch would probably still be fine but I would probably buy something thicker just for added durability.

As an alternative, and since this is a DIY forum, I feel I must mention that box springs are pretty simple structures and it would likely be pretty easy to just repair your current one.  They are mostly exactly what the name implies (a wooden box full of uncompressed springs covered with fabric).  I would encourage you to take the back covering off (its held on by staples most likely so its easy to re-attach) and assess the real damage.  Post pics of what you find here asking for advice if necessary but you are probably just looking at replacing a couple of broken 1x2.  Take a piece of the broken ones to the lumber yard with you to make sure you get the same kind of wood (or something with similar strength) and replace the full length of the broken piece (don't try to replace just a section of it).  If re-breaking is a concern, you can reinforce as necessary after the repair is completed or you could always add the plywood between the mattresses as well just for added assurance.