I cannot get the g in either capital or lower case to work. It just comes up with a square. Capital G (unicode: e0e1) and small g (e4f5) are what I am lacking.
U+e0e1 and U+e4f5 are characters from the Private Use Area of Unicode, which means they can contain anything the font designer chooses. There is no Unicode encoding for either Latin Small Letter g with Tilde, or Latin Capital Letter G with Tilde. To obtain these compositions it is necessary to use the Combining Tilde (U+0303) with the usual letter e.g.,
CTRL+SHIFT+U+0303 ENTER g
CTRL+SHIFT+U+0303 ENTER G
Note however that rendering for combining marks is often less elegant than that for a precomposed character. As you mention in the comment, given the right font (i.e., one with precomposed versions of these characters in the Private Use Area) the codes indicated will work as expected.