Going this route means you have to plan wisely for the graduate programs of interest to you, but it is quite feasible. Colleagues at MIT tell us that most graduate programs in engineering are as much interested in strong students with a broad-based science background from an excellent liberal arts college as they are in students with engineering degrees. Strong preparation in physics and mathematics is essential. Otherwise, the choice of MHC courses depends to some extent on what kind of engineering you're interested in. A chemistry major can lead to chemical engineering.
Computer science is needed for computer engineering.
Taking one or two engineering courses at UMass on the Five College Exchange is also useful. One caveat for the graduate school route is the issue of professional certification. Strangely, most graduate engineering programs aren't ABET-approved. Of course, not all engineering work requires the PE certification, although civil engineering does, and there are only two ABET-approved graduate programs in civil engineering. To register as a Professional Engineer in Massachusetts, for example, you need either an ABET accredited undergraduate degree plus four years of experience or an undergraduate degree in one of the sciences plus eight years of experience.