Actually, the rull academic formation to be either a priest or a brother begins after college. In the United States our education system is a little different from other countries, because of the way that we break down the academic career. Most countries do not have a Master's Degree.
In the USA we have a BA. Those students who are going on to a Master of Divinity or a Master of Arts in theology, which is different from a Masters of Theology, are required to have at least a minor in philosophy. These are specific courses in philosophy that are prerequisites for the study of theology and ministry.
The usual practice is that if you already have a BA or BS from an accredited university, you must complete the required courses in philosophy to begin the graduate studies in theology and ministry. Dioceses and religious communities are don't really make a big deal about where you get your undergraduate degree as long as your degree is accredited and you have the prerequisite philosophy courses. If you have not had any undergraduate theology courses, they will have some that they require, but the number is negligable. We're talking about a year of additional studies before entering graduate school for theology and ministry.
Also, a lot is going to depend on which Master's degree you're going for. If you're going for the Masters of Divinity (M.Div), the requirements are less stringent than if you're going for the Master of Arts in Theology (M.A.). If you're going to be a priest, the usual degree is the M.Div.
Because the M.Div is a practical degree, not a research degree, the undergraduate requirements are usually about 18 credits in philosophy and several courses in Christina doctrine, scripture and religious studies.
Contrary to popular belief, most students for the priesthood and brotherhood attend regular colleges for their undergraduate degrees, because most religious orders and most dioceses cannot afford to operate undergraduate colleges. There are still a few around. Most are regional seminaries called Minor Seminary. The term is actually misleading. The minor seminary was the high school seminary. When that disappeared, the college seminary became the minor seminary.
All that being said, there is no universal requirement in the USA that a man attend a college seminary before entering the major seminary (graduate school). Most religious communities and dioceses accept men who come with degrees from other colleges. As I said above, if you do not have the prerequisite undergraduate theology and philosophy courses, they will ask you to take them before admission to graduate studies.
These rules are only for the USA. Other countries are different, because their academic system is organized differently. In countries where there is no Master's Degree, which is most countries around the world, you simply enter university for a six-year program of studies toward a degree in a specific area. You would come out with a degree in Scripture, Moral Theology, Liturgy, Canon Law, Mystical Theology and so forth.
Br. JR, OSF :)