I use a variety of different base sizes, because I have variety in my unit sizes. But my infantry (excluding skirmishers) are all 200mm wide, with varying number of ranks.
My archers and smaller 'heavy infantry' units are 20 figures. I've also got one mixed unit of 'bill and bow' the same size. (I have larger heavy infantry units of 3 ranks (30 figures), and the pike units are bigger/deeper still).
So they are all 200x50mm. That's four 50mm square bases, with five figures per base. Though actually it's not, I have a central 100x50 (10 minis) flanked by 2 50x50 bases (5 minis each).
So yes, 5 figures on a 50x50. I'd recommend it! 50x60 obviously would be in the same ballpark.
If you went for 40x40 you'd only be able to fit 4 figures per base.
Now how squeezed in do you want them? I generally think for heavy infantry with polearms, they look MUCH BETTER squeezed in tight. Having too much space between them really spoils the look I think, and also takes up more room on the table! So some of my 20-strong units of heavy infantry actually have an extra figure or two, so they are 21 or 22.
With archers though they can get a bit cramped, they seem to take up more room, especially if in shooting and knocking poses.
So another option for a mixed 'bill & bow' unit is to have the central 100x50 (or two 50x50s) packed in a bit closer, perhaps 12 figures (mostly men-at-arms and billmen, perhaps an archer or two), and the flanking 50x50s can have 4 archers each.
I think around 20 is a good number of figures for this sort of unit.
Of course the Hail Caesar rules (and I'm sure most historical rulesets these days) aren't too bothered with how you base your figures, anything goes. It would makes sense for it to be broadly consistent for both sides.... but I don't think it really matters. Though if you are playing solo, so presumably doing both sides yourself, you can of course decide on your preferred unit size and basing method and keep it consistent!
Another thought with using 50x50 bases (or 50x60 or whatever) for mixed units - you don't have to be committed to just having mixed 'bill and bow' units. Switch the bases around to make up pure archer and pure bill units for games if you want. Paint up a few extra 50x50s to give you more options. You might find having every unit in the game more or less the same could be a bit boring, and someday will want to arrange the armies differently, separating the bills and bows, or having both separate and mixed units on the table together.