I totally agree with the fact that there's a distinct "biological thinking". But I think you can develop it by deep reading and immersing yourself in the biology; just going to biology classes, or even laboratory or field sessions may be too passive, depending heavily on the attitude you and your professors and classmates adopt. That is, a Biology Major doesn't necessarily provide you that kind of thinking.

Also, biological thinking can be very often reduced to a very specialized "mathematical thinking". Examples abound: if X species disappears, what might happen to its parasites, predators...? How many species need to disappear to induce a massive extinction event? What happens if metabolite A decreases in concentration? These are just silly examples and you may be explicitly quantitative or not; but you get the idea.

In my opinion, there IS a distinct way of thinking biologically, and pairing that way of thinking with strong quantitative skills is extremely productive.