What I have a difficulty to see is how PE can be negative.
Recall how Potential Energy (PE, here symbolized by $U$) is defined.
A potential energy $U$ is defined implicitly in terms of a force $\vec F$ via the equation:
$$
\vec F(\vec x) = -\vec\nabla U(\vec x)\;. \tag{1}
$$
Clearly, the potential energy is not defined uniquely via Eq. (1). Any other potential energy function that only differs from the first by a constant will result in the same physics.
For example, let:
$$
V(\vec x) = U(\vec x) + C\;,
$$
where $C$ is a constant (independent of spatial position).
Then we also have:
$$
\vec F = -\vec \nabla V\;.
$$
Therefore, you are free to add an arbitrary constant to your definition of potential energy in order to make the math easier.
Clearly, you can choose to make the potential energy at any given point positive, or negative, or zero, by definition. Thus, the sign of the potential energy is not necessarily of any significance.
Recall that the electrostatic force, due to a fixed particle of charge $Q$ at the origin, on a test particle of charge $q$ at position $\vec x$ is given by:
$$
\vec F = \frac{Qq\vec x}{4\pi\epsilon_0|\vec x|^3}\;.
$$
You are free to rewrite this force field $\vec F$ in terms of a potential energy field $U$, where
$$
U(\vec x) = \frac{Qq}{4\pi\epsilon_0|\vec x|}\;,
$$
which has the convenient property of approaching zero as $|\vec x|$ approaches infinity.
However, if you so please, you can also write the potential energy as:
$$
V = \frac{Qq}{4\pi\epsilon_0|\vec x|} + C\;,
$$
where $C$ is any constant value you would like it to be.
OK, so let's say we hate negative numbers for some reason and we want to try and make sure we never have to see any negative numbers by imposing a large (and physically irrelevant) C value.
Let's take $C=1000000$ so we can write:
$$
V = \frac{Qq}{4\pi\epsilon_0|\vec x|} + 1000000
$$
Classically, the total energy of a proton/electron system when the particles are very far apart and at rest is:
$$
E_1 = m_p c^2 + m_e c^2 + 1000000\;.
$$
Classically, the total of a proton/electron system when the particles are a distance $d$ apart and at rest is:
$$
E_2 = m_p c^2 + m_e c^2 + 1000000 - \frac{|e|^2}{4\pi\epsilon_0 d}\;.
$$
Even though the sign of each individual energy ($E_1$ and $E_2$) might be positive, the energy difference is still negative, regardless of the value of C:
$$
E_2 - E_1 = - \frac{|e|^2}{4\pi\epsilon_0 d}
$$
