1. Meaning of symmetry about the line $x=2$

If the graph of $y=f(x)$ is symmetric about the vertical line $x=2$, then points at equal horizontal distances from $x=2$ have the same $y$-value.

So, for any real number $x$, $f(2+x)=f(2-x).$

This is the standard condition for symmetry about the line $x=a$: $f(a+h)=f(a-h).$ Here $a=2$.

2. Check the options

• Option A: $f(x)=-f(-x)$

  This represents an odd-function type symmetry about the origin, not symmetry about $x=2$.

  So, A is incorrect.

• Option B: $f(2+x)=f(2-x)$

  This exactly matches the condition for symmetry about the line $x=2$.

  So, B is correct.

• Option C: $f(x)=f(-x)$

  This represents an even-function type symmetry about the $y$-axis, i.e. symmetry about $x=0$, not $x=2$.

  So, C is incorrect.

• Option D: $f(x+2)=f(x-2)$

  This does not represent reflection symmetry about $x=2$. It is a different relation and is not the required symmetry condition.

  So, D is incorrect.

3. Final answer

Therefore, the correct option is $\boxed{B}.$