I'm confused by a claim I've read in a textbook--that only boundary points can be support points. Here are the details.
Let $A$ be a nonempty subset of a topological vector space $X$. A point $x \in A$ is a support point of $A$ if there's a nonzero continuous linear functional on $X$ that is minimised over $A$ at $x$.
But according to this definition every point in $A$ is a support point: just consider the nonzero continuous linear function $f=1$. In particular, it's not the case that only boundary points are support points.
What have I misunderstood?
You have misunderstood linearity. The function $f = 1$ is not linear; it fails both additivity and scalar homogeneity. While a constant function will also minimise any $A$ at all points simultaneously, the only linear constant function is the $0$ functional, which is explicitly excluded.
Note also the other trivial case, where $X = \{0\}$, is included vacuously: there is no non-zero functional on $X = \{0\}$, so $A$ has no support points.