The following write-up is by David Tuller.  The answer to the last part is 84.

UNIT HEXAGONS: An equiangular hexagon has sides of  1, X, 3, 4, 5, and
6, going clockwise. What is X?  Hint: it isn't 2.  Related problem: An
equiangular hexagon has sides of 1, 2, 3, 4, 5, and 6.  What order are
these sides in?  Larger problem:  Equiangular hexagons with unit sides
are sorted by their area in unit triangles.  What is the largest area
not represented by a hexagon?

Theorem 1 - The sides of an equiangular hexagon can be (given in
clockwise order) A, B, C, D, E, F if and only if the following two
equations hold:

(1)   A + B = D + E
(2)   B + C = E + F

Proof - Just count how far you've travelled in the x and y directions as
you move around the hexagon.

Corollary 1 - If the sides of an equiangular hexagon are 1, X, 3, 4, 5,
6 given in clockwise order, then X=8.

Proof - Use equation (1) or (2).

Corollary 2 - If the sides of an equiangular hexagon are 1, 2, 3, 4, 5,
6 in some order then there are two possibilities up to isomorphism: 1,
6, 3, 2, 5, 4 and 1, 6, 2, 4, 3, 5.

Proof - WLOG, let B = 6. Then the only possibilities for A are 1, 2 or 3
since if A >= 4, then there are no choices of D, E to make equation (1)
true. The only ways to make equation (1) true then are (up to
interchanging D and E):

6 + 1 = 2 + 5
6 + 1 = 3 + 4
6 + 2 = 3 + 5
6 + 3 = 4 + 5

Then 1 must be adjacent to 6 so we can assume WLOG that A = 1. The rest
is easily broken into the two cases shown above.

Theorem 2 - If the sides of an equiangular hexagon are A, B, C, D, E, F
given in clockwise order, then the area of the hexagon is given by

(3)  (A + B + C)^2 - A^2 - C^2 - E^2 = B^2 + 2AC + 2BC + 2AB - E^2

Proof - An equilateral triangle of side n has area n^2 times the area of
a unit equilateral triangle. The theorem follows since the hexagon is an
equilateral triangle of side A + B + C minus triangles with sides A, C
and E.

Corollary 3 - N is the area of some equiangular hexagon if
1) N = 2 mod 4, N >=6
2) N is odd and N+2 is neither a prime nor a square of a prime
3) N is even and N+5 is neither a prime nor a square of a prime

Proof - Left to the reader. :-)

What's left to check at this point: 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 15,
17, ...

Although this doesn't prove that all but a finite number of Ns are the
area of some equiangular hexagon, it does at least show that "almost
all" Ns work and once an upper bound is known can be used to find all
such Ns.