What would happen if all protons in the Sun had a charge of 1.000001 times the elementary charge? The whole sun would be positively charge, but would it explode because of it? Or would it just expand a bit?
 A: The potential energy of a uniform sphere of charge $Q$ is:
$$ U_Q = \frac 3 5 \frac 1 {4\pi\epsilon_0} \frac{Q^2}R $$
while the gravitational binding energy is:
$$ U_G = -\frac 3 5 \frac{GM^2} R $$
(Both of these results are standard physics problems).
You want to add them:
$$ U(R) = U_Q(R) + U_G(R) $$
and find the minimum with respect to $R$. Of course, they have the same behavior vs $R$, so that is pointless. One would think it is minimized at $R=\infty$.
However, if you fix $R$, it would be more interesting to the find $Q$ for which:
$$ U_Q = U_G$$
Note that in this treatment, the Sun is approximated as a gas of non interacting Newtonian particles, for which there are only two outcomes: explosion or implosion, depending on the ratio of $Q/M$.
You can use:
$$ -\frac{d(U_Q+U_M)}{dR}$$
to compute the inward or outward pressure.
Without putting in any numbers, if $q_p = 1.000001|e|$, the Sun would be obliterated in fashion that make gamma ray bursts seem dull, as the electric force is $10^{39}$ times stronger than gravity.
A: The sun is made of a highly conductive plasma. The excess charge would move to the surface. The E field inside would be $0$.
The mutual force of repulsion would be huge. The surface charge would fly away at relativistic speed. The interior would not be affected by this.
Are you supposing the Sun suddenly changes from real physics to this? The Sun would start with excess charges distributed all through it. They would violently fly to the surface, and would no doubt rip apart the Sun as they did.

As JEB said, the E&M force is $10^{38}$ times stronger than gravity.
The mass of the Sun is about $2 \cdot 10^{30}$ kg of H. You can calculate the gravitational force by imaging that all that mass was at the center.
Suppose you put $10^{-8}$ kg = $10$ $\mu$g of protrons at the center. That would be enough to balance the Sun's gravitational attraction to a + charge on the surface.
You are talking about making $1$ million$^{th}$ the mass of the sun be extra + charges. That is about $10^{32}$ kg.
A: Not directly an answer to your question, but note that the sun is actually positively charged, by an amount that prevents electrons from escaping from the sun (electrons are effectively not bound by the gravitational field of the sun due their small mass/ high speeds and would practically all escape without an electric field holding them back). Once the charge is such that the outward flux of electrons and ions is the same, a steady state is reached and the charge stays the same.  See also https://www.aanda.org/articles/aa/pdf/2001/24/aah2649.pdf
