As Amit as commented, there is a mistake in the above expression, however- this is not the answer I got as I have reworked through the math. So the answer which I think is correct along with my work is the following:
we are looking to solve $$\arg \min_{x_1,x_2} p_1x_1+p_2x_2$$ subject to: $$\bar{U}\geq-(x_1-\delta_1)^2-(x_2-\delta _2)^2$$ where $\bar{U}\leq0$.
Step 1: Set up Lagrangian $$\mathcal{L}:p_1x_1+p_2x_2+\lambda[\bar{U}+(x_1-\delta_1)^2+(x_2-\delta _2)]$$
Step 2: Take FOCs and derive optimality condition $$(1)\ \ \ \ \ \ \ \frac{\partial\mathcal{L}}{\partial x_1}:p_1+2\lambda(x_1-\delta_1)=0$$ $$(2)\ \ \ \ \ \ \ \frac{\partial\mathcal{L}}{\partial x_2}: p_2+2\lambda(x_2-\delta_2)=0$$
Rearranging the above we get:
$$(1a)\ \ \ \ \ \ \ p_1=-2\lambda(x_1-\delta_1)$$ $$(2a)\ \ \ \ \ \ \ p_2=-2\lambda(x_2-\delta_2)$$
Dividing (2a) by (1a) we get our optimality condition (where MRS equals our price ratio).
$$\frac{p_2}{p_1}=\frac{x_2-\delta_2}{x_1-\delta_1}$$
Step 3: Derive "intermediate" demand equation
Rearranging our optimality condition we get:
$$\hat{x_2}=\delta_2+\frac{p_2}{p_1}(x_1-\delta_1)$$
Step 4: substitute intermediate demand equation into constraint to obtain hicksian demand
the following algebra below helps give our answer (apologies for the messiness).
$$\bar{U}=-(x_1-\delta_1)^2-(\hat{x_2}+\delta _2)^2$$
$$\bar{U}=-(x_1-\delta_1)^2-\left(\delta_2+\frac{p_2}{p_1}(x_1-\delta_1)-\delta _2\right)^2$$ $$\bar{U}=-(x_1-\delta_1)^2-\left(\frac{p_2}{p_1}(x_1-\delta_1)\right)^2$$ $$\bar{U}=-(x_1-\delta_1)^2-\left(\frac{p_2}{p_1}(x_1-\delta_1)\right)^2$$
$$\bar{U}=-(x_1-\delta_1)^2\left(1+\left(\frac{p_2}{p_1}\right)^2\right)$$
Heres the tricky part, recall that $\bar{U}\leq 0$ this allows us to do the following:
$$(x_1-\delta_1)^2=\frac{-\bar{U}}{1+\left(\frac{p_2}{p_1}\right)^2}$$ $$x_1-\delta_1=\left(\frac{-\bar{U}}{1+\left(\frac{p_2}{p_1}\right)^2}\right)^{\frac{1}{2}}$$
This is allowed because the term in the brackets is a positive number.
Moving over $\delta_1$ we have our hicksian demand for $x_1$:
$$x^h_1=\delta_1+\left(\frac{-\bar{U}}{1+\left(\frac{p_2}{p_1}\right)^2}\right)^{\frac{1}{2}}$$ by symmetry the hicksian demand for $x_2$ is:
$$x^h_2=\delta_2+\left(\frac{-\bar{U}}{1+\left(\frac{p_1}{p_2}\right)^2}\right)^{\frac{1}{2}}$$
I hope this helps.