Quadratic variation of the Ornstein-Uhlenbeck process

You cannot apply the formula

$$[G \bullet M]_t = \int_0^t G_s^2 \, d[M]_s \tag{1}$$

because the Ornstein-Uhlenbeck process $X$ is not of the form

$$X_t = (G \bullet B)_t,$$

but of the form $$X_t = (G_t \bullet B)_t$$ and -as your calculation show- we cannot expect that $(1)$ extends to this larger class of processes. The reason is, roughly, that $dt$-terms need a different compensation than $dB_t$-terms - and if you shift the multiplicative $dt$-term under the stochastic integral, then you pretend that it behaves, in some sense, like a $dB_t$-term ... but it doesn't.

The proper way is the following:

1. Define $$Y_t := \int_0^t e^{\alpha s} \, dB_s.$$ Calculate $[Y]_t$ (that you can do using $(1)$.)
2. Apply Itô's formula to find the stochastic differential $$d(X_t^2) = \sigma^2 d(e^{-2\alpha t} Y_t^2).$$
3. The $dt$-term of the stochastic differential $d(X_t^2)$, obtained in step 2, equals the quadratic variation $[X]_t$.