You definitely do not need a maters degree to go into aerospace engineering
I go to Cornell and was actually in the Mechanical Engineering program for a while, and know a lot of people in the program. I know a lot of people who with only bachelors have gone on to Boeing, Lockheed, Spacex etc. Yes the Masters helps, but it is by no means necessary. If you get hired with a masters, usually you get a little higher starting pay and get slightly more interesting projects, but you can also work your way up to those projects and pay if hired as a BS.
Also, as a side note, if you do decide you want a masters, Cornell has a one-year masters of engineering program that can be done in a single additional semester if you start it the spring of your senior year, meaning you get a bachelors and masters in 4.5 years. This program is pretty much auto-admit for any Cornell engineers (even if you do it in 5 years), so if you decide later on that a masters is what you want, you don't need to go through the hassle of applying to a bunch of places and hoping you get in somewhere.