Firearms are extremely simple machines; basically single-piston engines without the complication of a driveshaft
. Their manufacture is what's complicated. An engineering degree, honestly, is overkill for mere
gun design; engineering as we know it now didn't exist when most designs we use today were perfected, and hundreds of self-taught inventors and machinists design and build excellent weapons everyday. Check any gun building forum for examples. But making designs that can be mass-produced cheaply and effectively while satisfying customer needs is what engineering is all about.
That said, a Mechanical Engineering degree is about the broadest type of engineering qualification there is, and can get you entry into any field requiring design, manufacture, optimization, or maintenance (pretty much everything). I would think that if you wish to end up designing firearms (or ultimately managing those who do), you'll need a good background in mechanical design (from the degree), machine work (from extra-curricular experience in a shop), as well as knowledge in the field of
marketing, so you can effectively pitch your ideas and contributions. Ironically, those attributes (plus some courses in business management) are basically what you need to strike out on your own.
If you go to work in any organization where you will be responsible only for part of the process, you will find "design" and "engineering" are two different things. In a field as competitive and non-innovative as firearms (by which I mean there are few new products that can't be directly compared to old ones), I imagine the marketing guys are the primary drivers of design goals, while the technical guys' job is to figure out how to produce that design fast and cheap enough. Try to determine if you have a preference for either (there's plenty of creativity in both, but different kinds, and there is much wider need for the latter).
With the massive sustained increase in demand for guns (and ammo), there's a lot of shops trying to tool up for expanded production right now. If you can convince them you're qualified to help with that (degree or otherwise), that's your "in". Good luck.
TCB
--BS Aerospace Engineering
PS-most mechanically minded guys are drawn to firearms, but I think it's more because they are the last remaining purely mechanical system around anymore, than because the design challenges themselves are anything particularly special.