ZB, ceiling/floor transmission is something that's pretty difficult to address without construction, simply due to the way a room is built and the flanking of sound and noise. Soundproofing is essentially an all-or-nothing, full-room construction project.
With that said, eggcrate foam could diminish a very slight amount of sound just by existing as more mass than was originally there, but isn't going to give you the results you're after for your dollar or for this kind of task. Again, without construction, to even approach the deadening you're after, a denser material like closed-cell foam sheeting such as neoprene (eggcrate is open-cell, just as an FYI) would be needed, or a few inches of fiberglass batting insulation.
EDIT: Sorry, to address your "some better than none" question directly, more mass will help to a degree, but there isn't a concrete correlation where if you add X, you'll get Y. In absolute terms more mass helps, but with so many other things in play like the building's construction, material costs, effort to add materials, your own personal idea of how much "noise reduction" is acceptable, etc., it always ends up being a case-by-case situation where you have to decide if what goes into adding something is worth the result.
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