So, I tripped over this info: http://sheldonbrown.com/cribsheet-spacing.html
Looking over the info, it appears that I probably have either a Sun Tour "Ultra" 7 speed, or a Sun Tour "Microdrive" (the only 7-speed Sun Tour freewheels).

While the Microdrive appears to be fairly unique in its specs, the Ultra 7 seems like it has nearly identical cog spacing to the Shimano HG, and differs only in overall width. If I do indeed have the Ultra 7, can I assume that a Shimano HG freewheel would be compatible with the old shifters? Or would the extra width cause an issue?