There are two different versions of the Brownie Hawkeye. The older version, with metal winding knob and "long" on the bulb exposure toggle, will take unaltered 120 supply rolls (I'm told by those who have both versions). It also has a glass lens, making it the more desirable version. The newer version, with plastic winding knob and "L" on the bulb toggle, will take a *trimmed* 120 spool -- spool length is apparently not a problem, but the flange diameter must be reduced to that of a 620, which is easily done with a nail clipper or curved nail scissors, by cutting even with the surface of the paper on the roll.
Kodak did practice lock-in from the day they brought out 620 film. Their avowed purpose was to allow the design of slimmer, lighter cameras with the same negative sizes as corresponding 120 models, but in fact, it was mostly to force people to buy 620 film (silly of them, they should already have known from previous 116/616 experience that other companies would copy the 620 format in a matter of months if the cameras sold well). Many cameras from the heyday of 620 (Medalist, Kodak Reflex II, Tourist II, Vigilant 620) were tight enough as sold that they won't accept even modern plastic 120 spools with a flange trim, though some can be converted to take a 120 supply without much trouble.
In fact, though, the tightest 620 I own is a Savoy 620 -- it's tight enough to cause drag and film stretching even with original 620 spools. Don't recall exactly who made it, but it's a short step cheaper even than a Brownie Hawkeye; curved film plane, front glass meniscus lens, and no bulb setting. The back won't even slide on with a 120 in the supply -- the 0.1 inch difference in length is enough to bollix it up. Same is true of my Ansco Pioneer 620, but I can see in that camera where a little careful work with a Dremel will open things enough to use 120 at least on the supply side.