A speaker physics question.

[voice of the theater]

Yep it is fun, now to put them back where they sound best, in the Heathkits.

I'm not sure I understand exactly what you're describing as a "boxy" sound, but I think I have a general idea of what you're referring to. It may be something that can be fixed by things like (or a combination of):
1) speaker placement i.e. the difference between how they sound on the floor compared to how they sound sitting on stands/elevated off the floor which also puts the drivers at ear level. Keep in mind you can experiment with this before "committing to building speaker stands/risers" by propping them up on blocks (as long as they're solidly supporting all four sides of the bottoms of the speakers of course). You may very well be surprised at the difference in sound of the speakers being elevated compared to sitting on the floor (the only way to know which position if any gives you less or no boxy sound is to try both). Another possible factor related to speaker placement may be how near or far you have them from the corners of your room and/or how close you have them to the wall that's directly in back of them. Sometimes moving them just a few inches closer to the "back wall" or farther from that wall (or just a few inches closer to the corners of the room or farther) can make a surprising difference in sound.
2) EQ You may find with a little trial and error with an equalizer you can eliminate what you're referring to as a "boxy" sound. In my experience, most vintage Altecs I've owned don't require that you add much if any midrange EQ, but you may find an equalizer setting (possibly just adding some of the higher frequencies and probably the lower bass too) may "open up" your sound--and possibly if EQ experimentation is combined with speaker placement/elevation experimentation.
3) Crossovers I think you've mentioned that you have two sets of crossovers (an original set and a GPA set) and I'm aware you have two different models of 604. Maybe a little experimentation there might help as well--for example trying the original crossover with one 604 and the GPA with the other and vice versa or trying both old and both new and seeing which fit your system (referring to your entire system--drivers, cabs, amp, EQ, source, acoustics/room, etc.) the best. Keep in mind, the original crossovers probably need new caps if you want to give them a fair audition/comparison. Also, an active crossover might be worth experimenting with if these other variables aren't the "culprit" or as I mentioned, any combination of these three things (for starters). Just thinking out loud, for whatever that's worth......