I've been offered to work under one of my professors at my current University to get a PhD. I'm currently at a relatively large University in the United States studying Metallurgical engineering. There are various reasons why I think it would be advantageous to stay based on my situation, but I'm also aware that based on my grades and experience I could go to just about anything school in the country. The biggest thing I'm concerned about is the rumor that doing a PhD at the same school you do your undergraduate degree is highly frowned upon in the hiring process. I would like to know to what extent this is true and why.
You need to clarify what your career goals are.
If you want to stay in academia, this is sometimes frowned upon, at least in Europe, although it's not really unusual. It can also bring you huge advantages - a boss you know and who promotes you during your Ph.D. and afterwards (publications, conferences, projects, scholarships, teaching experience), so sometimes it's a great choice even though outsiders might frown upon it. What matters in academia are publications and projects anyways and nobody would reject your candidacy for a postdoc or a professorship just because you didn't switch your uni after your first degree if the publications and projects are ok. (But this is a topic for the academia exchange more than workplace).
If you want to work outside academia, it's improbable that anybody will care.
PhD at the same school you do your undergraduate degree is highly frowned upon in the hiring process
It's true that universities do not generally hire former alumni, but there are exceptions. Good research will likely out-weigh this concern.
I think it would be advantageous to stay based on my situation, but I'm also aware that based on my grades and experience I could go to just about anything school in the country.
This will have to be a personal choice. You've done a pretty good job of arguing why to stay. Here are some questions that may help you decide what's right for you.
1) If you can truly go anywhere in the country, you should highly consider going to the top-ranked school. While it's true PhDs are generally graded by quality of research, having MIT, Harvard, Yale, Princeton will likely have significant sway with hiring committees.
2) Figure out which university will give you the best fellowships/scholarships. Essentially this is the same game as being an exceptional high-school student and getting a free ride + room and board.
3) Going to another university will give you twice the network. When you graduate, you'll have twice the number of professors to reach out to.
4) Currently, there are far more PhD holders than professorships available. If you want to be a professor, realize you will be competing with very smart, very driven people from all over the world. This is true not just for top-ranked universities but for most, if not all, tenure-track positions.