My formulas are:

y=(x+a)^2

x=(-y+a)^2

I would like to solve for a where they touch (exact solution).

Here is a graph demonstrating what I mean:

As you can see, I have attempted to solve for a by making them closer and closer.

I think 1.100917368760402698 ≤ a ≤ 1.100917368760402699, but floating point error may have reduced the accuracy of that range, but it should be near those numbers at least.

There will also be a negative solution of a, which will be exactly negative I believe.

You can find a solution by expressing both graphs as functions of x. For the blue graph, this is done by splitting it in two parts g_1 and g_2. Finding values for a where they touch can be found by solving f(x) = g_1(x) and f'(x)=g_1'(x) simultaneously and the same for g_2. The answer does not look very nice, it has some nested roots, but it can be found by a solving a quartic equation. Indeed the positive value for a is approximated as 1.1009 by WolframAlpha.