What should Spec Z[\sqrt{D}] x_{F_1} Spec \bar{F_1} be? What should be $\text{Spec } \mathbb{Z}[\sqrt{D}] \times_{\mathbb{F}_1} \text{Spec } \overline{\mathbb{F}}_{1}$?
Sure, there's more than one definition.
I'm looking for any answer that uses at least one definition of scheme over $\mathbb{F}_1$.
This really is more a question of opinion.
What do you think this should be?
Some monoid that has something to do with $\text{Spec }\mathbb{Z}[\sqrt{D}][\mathbb{Q}/\mathbb{Z}]$ would be my guess (where the second brackets mean group ring).
This interests me from the point of view that, say, hyperelliptic curves over a finite field come (geometrically) from the group scheme of a quadratic extension of $\overline{\mathbb{F}}_p [t]$. In this case the frobenius acts on ideal classes, and satisfies a quadratic equation.
But, from what I understand, the natural analogue of frobenius in the arithmetic case, is like taking any positive power, and taking limits to 0 (or something of the sort).
Would this satisfy some kind of equation on, say, $\text{Pic(Spec } \mathbb{Z}[\sqrt{D}] \times_{\mathbb{F}_1} \text{Spec } \overline{\mathbb{F}}_{1}\text{)}$?
(for whatever definition of Pic that should be natural here)
I've searched for information on $\mathbb{F}_1$, but most just talk about making $\text{Spec }\mathbb{Z}$ into a curve, getting zeta functions to be Riemann's, etc.
Instead, I want to ask questions that are not just about proving the Riemann hypothesis, like the one above.
 A: Sorry I didn't reply before, I somehow didn't read the question till now.
I think your question is a bit misguided. The main problem I see with it is: what is $\text{Spec} \mathbb{Z}[\sqrt{D}]$ over F1? If you think of it as the M_0 scheme given by $\mathbb{Z}[\sqrt{D}]$ as a multiplicative monoid, then it is something huge (since that monoid is not even finitely generated, for starters), so none of the current notions can effectively deal with it (so far one can mostly only control schemes of finite type). If you want to make better sense of the question, one should ask: is it possible to find an algebra $A$ over F1 such that its base extension to $\mathbb{Z}$ gives $A\otimes_{\mathbb{F}_1} \mathbb{Z} = \mathbb{Z}[\sqrt{D}]$? 
If you find an answer to this question, say in CC setting where it boils down to finding a monoid (with zero) such that the reduced semigroup ring gives you back your original ring, 
then you might consider an approximation to your problem (the tensor with the abelian closure of F1) by taking some kind of multivariable-cyclotomic completion of your monoid, as in the papers by Manin and Marcolli.
