I cannot believe how many keyword searches I tried before I found this! But when I googled "National Lemon Squeezer"

- this came up:
c. 1903
https://books.google.com/books?id=M...v=onepage&q="national lemon squeezer"&f=false
Great research!
Now that we see what the missing parts look like, I'm going to out on a limb (somewhat) and suggest that this product was not a commercial success - possibly for many of the reasons I mentioned previously in the research. (And that might also explain why it was so difficult to track down -- if it were more successful, presumably the research would have been easier?)
Anyway - the missing rod (held by small thumbscrew) looks like it's there to help position the glass or cup, so juice doesn't go on the counter.
Still not clear on whether the handle is forced upwards to squeeze the lemon, or pulled downward towards the operator.
My guess is it's pushed upwards (as shown), but that might be incorrect.
Would be interesting to see it in action. I'm still not convined it's a "press", per se. Maybe a rotational "grinder"? At least that way, it would not put undo force on the latch (which I believe would break quickly.)
Finally, once you've squeezed the lemon and removed the glass, I think this thing would still drip all over the place and make a mess.
Even if you removed the lemon carcass.
For that reason, and others, I would be a little surprised to hear if very many of these were sold.
The other fruit juicers on the market at the time would not have suffered any of these deficiencies and "should" have captured the bulk of the market share.
Jeff might well have one of only a very few proof-of-concept units ever constructed??