Once we have @ThorGirl's derivation for $ SE(\hat{\beta_1})^2 $ we can use that to derive the Standard Error for $ \hat{\beta_0} $ i.e, $ SE(\hat{\beta_0})^2 $
Note: If you are looking for a step-by-step explanation of @ThorGirl's
answer take a look at this video.
We are going to use the following assumptions / observations:
1) Each output $y_i$ is assumed to have this form: $ \beta_0 + \beta_1x_i + \epsilon_i$
2) $\overline{y} = \frac{1}{n}\sum_{i=1}^{n} y_i$
2) The error terms $ \epsilon_i $ are uncorrelated with constant variance $ \sigma^2$
3) The $y_i$'s, $\hat{\beta}_0$ and $\hat{\beta}_1$ are random variables.
4) The $x_i$'s are known constants. The $\beta_0$ and $\beta_1$ are not necessarily known but they are also constants, they are not random variables.
During the derivation we are going to use the following properties of variance. Here $a$ is a constant and $x$ and $y$ are random variables.
1) $Var(a+x) = Var(x) $
2) $Var(ax) = a^2Var(x) $
3) $ Var(x+y) = Var(x) + Var(y) + 2 Cov(a,b) $
We start with the definition of $\hat{\beta}_0$ :
$$ \overline{y} - \hat{\beta}_1 \overline{x } = \hat{\beta}_0 $$
Take the variance of it to find the standard error:
$$ Var( \overline{y} - \hat{\beta}_1 \overline{x }) $$
At this point if $\overline{y}$ and $\hat{\beta}_1$ are uncorrelated we can take each term's variance. This uncorrelation is what @Flowsnake's answer refers to. That uncorrelation is not being proven here.
$$ Var( \overline{y}) - Var(\hat{\beta}_1 \overline{x }) $$
The $\overline{x}$ is constant here. It is generally assumed that the predictors, inputs $x$'es are known. Take that out of the variance.
$$ Var( \overline{y}) - \overline{x}^2 Var(\hat{\beta}_1) $$
@ThorGirl's answer has the value for $Var(\hat{\beta}_1)$ substitute that:
$$ Var( \overline{y}) - \overline{x}^2 \frac{\sigma^2}{ \sum_{i=1}^ {n} (x_i-\overline{x})^2 } $$
Now going to focus on the $Var(\overline{y})$. That is not a constant. Write its open form:
$$ Var( \frac{1}{n} \sum_{i=1}^{n}y_i) - \overline{x}^2 \frac{\sigma^2}{ \sum_{i=1}^ {n} (x_i-\overline{x})^2 } $$
We also know the $y_i$ terms by the definition of the model substitute that too.
$$ Var( \frac{1}{n} \sum_{i=1}^{n} (\beta_0 + \beta_1x_i + \epsilon_i)) - \overline{x}^2 \frac{\sigma^2}{ \sum_{i=1}^ {n} (x_i-\overline{x})^2 } $$
Take the $1/n$ out of the variance.
$$ \frac{1}{n^2} Var( \sum_{i=1}^{n} (\beta_0 + \beta_1x_i + \epsilon_i)) - \overline{x}^2 \frac{\sigma^2}{ \sum_{i=1}^ {n} (x_i-\overline{x})^2 } $$
The $\epsilon_i$ 's are uncorrelated random variables. With that, the variance of sum of random variables becomes the sum of their individual variances.
$$ \frac{1}{n^2} \sum_{i=1}^{n} Var(\beta_0 + \beta_1x_i + \epsilon_i) - \overline{x}^2 \frac{\sigma^2}{ \sum_{i=1}^ {n} (x_i-\overline{x})^2 } $$
The constants ($\beta_0 + \beta_1x_i$) in the variance term can be ignored. $Var(a + x) = Var(x) $
$$ \frac{1}{n^2} \sum_{i=1}^{n} Var(\epsilon_i) - \overline{x}^2 \frac{\sigma^2}{ \sum_{i=1}^ {n} (x_i-\overline{x})^2 } $$
By our assumptions, the variance of each $\epsilon_i$ random variable is constant and is $\sigma^2$ . Replace that.
$$ \frac{1}{n^2} \sum_{i=1}^{n} \sigma^2 - \overline{x}^2 \frac{\sigma^2}{ \sum_{i=1}^ {n} (x_i-\overline{x})^2 } $$
Summing $\sigma^2$ $n$ times. Simplify that and cancel $n$ from the nominator and denominator
$$ \frac{1}{n^2} n \sigma^2 - \overline{x}^2 \frac{\sigma^2}{ \sum_{i=1}^ {n} (x_i-\overline{x})^2 } $$
$$ \frac{1}{n} \sigma^2 - \overline{x}^2 \frac{\sigma^2}{ \sum_{i=1}^ {n} (x_i-\overline{x})^2 } $$
Factor out the $\sigma^2 $
$$ \sigma^2 \left [ \frac{1}{n} - \frac{\overline{x}^2}{ \sum_{i=1}^ {n} (x_i-\overline{x})^2 } \right ] $$
We found the $SE(\hat{\beta_0})^2$
