Is this an energy or power signal?
As a complement to Matt's answer, on the intuition: $u[n]$ has value $1$ from $n=0$ on. So basically its energy will increase for ever, because it keeps adding ones for $n\ge0$. Then, you build another signal $x[n]$ that grows way faster because you multiply it with the exponential term $4^n$.
Therefore, only at the common sense level, one cannot expect it to be an energy signal. It is not feasible.
And then, goes the proof that the sum of the series actually diverges. I would suggest to use the simplest proofs, to avoid complexities, and formulae that are not valid (sum of geometric series for $a \ge1$).
$$\sum_{n=-N}^N  |4^nu[n]|^2 \ge \sum_{n=0}^N  4^{2n}|u[n]|^2\ge \sum_{n=0}^N  |u[n]|^2 \ge N+1$$
which diverges. Hence the signal is not energy. Something similar can be used for power.
Of course, as Matt said, if the  exponential term becomes $4^{-n}$, this geometrical series then decreases fast to zero, and wins over the mere sum of ones in $u[n]$.