jmc

The graph of the two equations is shown below:



We first find the $x$ values at which the two equations intersect. When $x\ge 0$, $y=|x|=x$. Plugging this into the second equation to eliminate $y$, we get $x=-x^2+\frac{35}{4}\Rightarrow x^2+x-\frac{35}{4}=0$. Factoring the left hand side gives $\left(x+\frac{7}{2}\right)\left(x-\frac{5}{2}\right)=0$, so $x=2.5$ (since we stated that $x$ was non-negative). By symmetry, the $x$ value of the left intersection is $x=-2.5$. So we just have to consider the integer $x$ values between these two bounds and find all integer $y$ values that make the point $(x,y)$ fall inside the region.

For $x=-2$, the value of $y=|x|$ is $y=2$ and the value of $y=-x^2+\frac{35}{4}$ is $y=\frac{19}{4}=4.75$, so all $y$ values between 2 and 4 inclusive work, for a total of 3 points. For $x=-1$, the value of $y=|x|$ is $y=1$ and the value of $y=-x^2+\frac{35}{4}$ is $y=\frac{31}{4}=7.75$, so all $y$ values between 1 and 7 inclusive work, for a total of 7 points. For $x=0$, the value of $y=|x|$ is $y=0$ and the value of $y=-x^2+\frac{35}{4}$ is $y=\frac{35}{4}=8.75$, so all $y$ values between 0 and 8 inclusive work, for a total of 9 points. By symmetry, when $x=1$, there are 7 points that work, and when $x=2$, there are 3 points that work.

In total, there are $3+7+9+7+3=\boxed{29}$ lattice points in the region or on the boundary.