It looks like you have to be mores specific about what you mean about when $y=0$
At $(-1,0)$ you have a vertical tangent line $x=-1$ with undefined slope.
At $(0,0)$ you are correct there is not a uniquely defined tangent line. However you can make a uniquely defined tangent line of you parameterize the curve. That is both $x$ and $y$ would be functions of some parameter for example $t$ then the curve will be drawn $(x(t),y(t))$.
In this way of looking at the curve, the self intersection will be two unique $t$-values and you can draw a distinct tangent line for each one and it looks like the slope will actually be defined in that case, but you are adding extra information with the parameterization.
Edit Also I might add that not just any parameterization will work you'd need to pick a differentiable one, that is $\frac{dx}{dt}$ and $\frac{dy}{dt}$ both exist and are not simultaneously zero. This forces no sharp turns along the path you follow. So at the offending point $(0,0)$ you'd need to make sure it follows to the opposite quadrant as $t$ gets larger.
Just to complete what I was saying above I found a differentiable parameterization of the curve
$$
\begin{split}
x&=\tan^2 t -1\\
y&=\tan^3 t- \tan t
\end{split}
$$
Which has derivatives
$$
\begin{split}
\frac{dx}{dt}&=2\tan t\sec^2 t \\
\frac{dy}{dt}&=3\tan^2 t \sec^2 t- \sec^2 t
\end{split}
$$
On the interval $-\frac{\pi}{2} < t < \frac{\pi}{2}$
And the $(0,0)$ point on the graph occurs at the $t$-values $t=-\frac{\pi}{4}$ and $t=\frac{\pi}{4}$
$$
\begin{split}
\frac{dx}{dt}|_{t=\frac{\pi}{4}}&=4 \\
\frac{dy}{dt}|_{t=\frac{\pi}{4}}&=4\\
\frac{dy}{dx}|_{t=\frac{\pi}{4}}&=\frac{dy/dy|_{t=\frac{\pi}{4}}}{dx/dt|_{t=\frac{\pi}{4}}}=1
\end{split}
$$
$$
\begin{split}
\frac{dx}{dt}|_{t=-\frac{\pi}{4}}&=-4 \\
\frac{dy}{dt}|_{t=-\frac{\pi}{4}}&=4\\
\frac{dy}{dx}|_{t=-\frac{\pi}{4}}&=\frac{dy/dy|_{t=-\frac{\pi}{4}}}{dx/dt|_{t=-\frac{\pi}{4}}}=-1
\end{split}
$$
I'm fairly sure this curve is well-studied, but don't know the name of it and just felt like deriving something like this myself :)
